The concept of a limit is the foundational idea of analysis — the mechanism by which infinite processes produce definite answers. Derivatives are limits of difference quotients, integrals are limits of Riemann sums, [power series](/page/Power%20Series) are limits of partial sums, and solutions to differential equations are limits of iterative approximations. Without a precise definition of limit, none of these constructions can be made rigorous, and the paradoxes of the infinite (Zeno's paradox, the divergence of the harmonic series, the existence of continuous nowhere-differentiable [functions](/page/Function)) cannot be resolved. This page develops the concept of limit from the ground up: the $\varepsilon$-$N$ definition for [sequences](/page/Sequence) in $\mathbb{R}$, the fundamental convergence theorems (monotone convergence, squeeze theorem, Bolzano-Weierstrass), the extension to metric and [topological spaces](/page/Topology), the $\varepsilon$-$\delta$ definition for functions, [Cauchy sequences](/page/Cauchy%20Sequence) and completeness, and the delicate question of when limits can be interchanged. [motivation] ## Motivation ### The Problem of the Infinite Mathematics constantly encounters infinite processes that produce finite answers. The decimal expansion $0.333\ldots = 1/3$ asserts that an infinite sum of fractions equals a rational number. The area under a curve is obtained by summing infinitely many infinitesimal rectangles. The derivative $f'(x)$ is the value of the ratio $(f(x+h) - f(x))/h$ "at $h = 0$" — but dividing by zero is undefined. In each case, the resolution is the same: the infinite process does not reach its destination in finitely many steps, but it *approaches* it with arbitrary precision. Making "approaches with arbitrary precision" into a rigorous mathematical statement is the purpose of the limit concept. ### Why Intuition Fails The intuitive notion — "$a_n$ gets closer and closer to $L$" — is insufficient. The sequence $a_n = (-1)^n / n$ oscillates around $0$ and converges to $0$, even though it is not monotonically approaching. The sequence $a_n = 1 + (-1)^n / n$ gets closer and closer to $1$ at every step, yet so does $b_n = 2 - 1/n$ — which converges to $2$, not $1$. "Getting closer" does not determine the limit; what matters is *staying within any prescribed tolerance forever after some point*. This is the insight that Cauchy and Weierstrass formalised in the 19th century: a sequence converges to $L$ if for every $\varepsilon > 0$ (no matter how small), the terms $a_n$ are eventually within $\varepsilon$ of $L$. ### From Real Numbers to Abstract Spaces The $\varepsilon$-$N$ definition uses the absolute value $|a_n - L|$ to measure distance. In $\mathbb{R}^n$, the Euclidean distance replaces the absolute value. In a general metric space, an arbitrary distance function $d$ plays this role. In a [topological space](/page/Topology), where no distance may exist, convergence is defined by requiring that the sequence eventually enters every [open set](/page/Open%20Set) containing the limit. Each generalisation preserves the essential structure — "eventually within any prescribed neighbourhood" — while broadening the scope. [/motivation] ## Definition The limit of a sequence is the value that the terms approach as the index grows without bound. The precision of "approach" is controlled by the $\varepsilon$-$N$ quantifier structure: *for every tolerance $\varepsilon$, there is a stage $N$ beyond which all terms are within $\varepsilon$ of the limit*. [definition:Limit Of A Sequence In R] Let $\{a_n\}_{n=1}^\infty$ be a sequence of real numbers. We say $\{a_n\}$ **converges** to $L \in \mathbb{R}$, and write $\lim_{n \to \infty} a_n = L$ or $a_n \to L$, if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that \begin{align*} n \ge N \implies |a_n - L| < \varepsilon. \end{align*} A sequence that converges to some $L \in \mathbb{R}$ is called **convergent**. A sequence that does not converge is **divergent**. [/definition] The definition has three quantifiers: $\forall \varepsilon > 0$, $\exists N$, $\forall n \ge N$. The order matters: $N$ is allowed to depend on $\varepsilon$ (small tolerances require waiting longer), but once $N$ is chosen, the bound $|a_n - L| < \varepsilon$ must hold for *all* $n \ge N$, not just some. This "for all sufficiently large $n$" structure is what distinguishes convergence from the weaker condition "infinitely many terms are close to $L$" (which describes limit points of subsequences, not limits). A convergent sequence has exactly one limit: [quotetheorem:625] The proof in [metric spaces](/page/Metric%20Space) is a one-line application of the triangle inequality: $d(x, y) \le d(x, x_n) + d(x_n, y) < \varepsilon/2 + \varepsilon/2 = \varepsilon$ for large $n$, so $d(x, y) < \varepsilon$ for all $\varepsilon > 0$, giving $d(x,y) = 0$. In general topological spaces, the Hausdorff condition (disjoint open separating [sets](/page/Set)) replaces the triangle inequality. In non-Hausdorff spaces, limits need not be unique — in the indiscrete topology, every sequence converges to every point. [example:Basic Convergent Sequences] The sequence $a_n = 1/n$ converges to $0$: given $\varepsilon > 0$, choose $N > 1/\varepsilon$ (which exists by the Archimedean property of $\mathbb{R}$), and then $n \ge N$ implies $|1/n - 0| = 1/n \le 1/N < \varepsilon$. The sequence $a_n = (n+1)/(2n+3)$ converges to $1/2$: we compute $|a_n - 1/2| = |(2(n+1) - (2n+3))/(2(2n+3))| = 1/(2(2n+3))$. Given $\varepsilon > 0$, choose $N$ with $1/(2(2N+3)) < \varepsilon$; then $n \ge N$ gives $|a_n - 1/2| < \varepsilon$. The sequence $a_n = (-1)^n$ diverges: it oscillates between $-1$ and $1$, so $|a_n - L| \ge 1$ for infinitely many $n$ regardless of the choice of $L$ (specifically, if $L \ge 0$ then $|a_n - L| \ge 1$ for odd $n$; if $L < 0$ then $|a_n - L| \ge 1$ for even $n$). [/example] ## Algebra of Limits The practical computation of limits relies on algebraic rules that allow limits of complicated sequences to be computed from limits of simpler ones. These rules follow from the $\varepsilon$-$N$ definition by triangle-inequality and estimation arguments. ### Arithmetic Operations If $a_n \to L$ and $b_n \to M$, the basic algebraic rules are: $a_n + b_n \to L + M$ (choose $N$ so that $|a_n - L| < \varepsilon/2$ and $|b_n - M| < \varepsilon/2$, then $|(a_n + b_n) - (L+M)| < \varepsilon$); $a_n \cdot b_n \to L \cdot M$ (use the identity $a_n b_n - LM = a_n(b_n - M) + M(a_n - L)$ and the fact that convergent sequences are bounded); and $a_n / b_n \to L/M$ provided $M \neq 0$ (first show $1/b_n \to 1/M$, then use the product rule). These rules extend to finite linear combinations: if $a_n^{(1)} \to L_1, \ldots, a_n^{(k)} \to L_k$ and $\alpha_1, \ldots, \alpha_k \in \mathbb{R}$, then $\sum_{i=1}^k \alpha_i a_n^{(i)} \to \sum_{i=1}^k \alpha_i L_i$. This is the statement that the limit is a **linear functional** on the space of convergent sequences. ### Order Properties The limit respects the ordering of $\mathbb{R}$. If $a_n \le b_n$ for all $n$ (or all sufficiently large $n$) and both sequences converge, then $\lim a_n \le \lim b_n$. The inequality is not strict: $1/n > 0$ for all $n$ but $\lim (1/n) = 0$. This is a common source of error — limits preserve non-strict inequalities but can collapse strict ones. The most powerful consequence of the order properties is the squeeze theorem: [quotetheorem:627] The squeeze theorem is often the only available tool when a sequence has no closed-form expression. The typical strategy is to bound the sequence between two sequences whose limits can be computed explicitly. [example:Squeeze Theorem Application] We compute $\lim_{n \to \infty} \frac{\sin(n)}{n}$. Since $-1 \le \sin(n) \le 1$ for all $n$, we have $-1/n \le \sin(n)/n \le 1/n$. Both $-1/n \to 0$ and $1/n \to 0$, so by the [Squeeze Theorem](/theorems/627), $\sin(n)/n \to 0$. [/example] ## Monotone and Bounded Sequences The algebra of limits computes the limit of a sequence whose convergence is already known. The deeper question is: *when does a sequence converge at all?* For sequences in $\mathbb{R}$, the completeness of the real numbers provides two fundamental convergence criteria: monotone bounded sequences converge (the Monotone Convergence Theorem), and bounded sequences have convergent subsequences (Bolzano-Weierstrass). ### The Monotone Convergence Theorem A sequence that is both monotone and bounded is forced to converge — the limit is the supremum (for increasing sequences) or infimum (for decreasing sequences). This is a direct consequence of the least upper bound property of $\mathbb{R}$. [quotetheorem:626] The power of this result is that it guarantees convergence without requiring knowledge of the limit. The proof uses completeness in the most direct way possible: the supremum exists (by the least upper bound property), and monotonicity forces the terms to approach it. This theorem is the foundation of many constructions in analysis: the definition of $e = \lim (1 + 1/n)^n$ (the sequence is increasing and bounded above by $3$), the construction of the [Riemann integral](/page/Riemann%20Integral) (upper and lower sums are monotone and bounded), and the proof that every bounded increasing sequence of sets has a limit in measure theory. [example:Computing E Via Monotone Convergence] Define $a_n = (1 + 1/n)^n$. To show $\{a_n\}$ is increasing, apply the AM-GM inequality or the binomial theorem: the binomial expansion shows that each term in the sum increases with $n$, and new positive terms are added. To show $\{a_n\}$ is bounded above, use the binomial theorem and bound: \begin{align*} (1 + 1/n)^n = \sum_{k=0}^n \binom{n}{k} \frac{1}{n^k} \le \sum_{k=0}^n \frac{1}{k!} \le 1 + \sum_{k=1}^n \frac{1}{2^{k-1}} < 3. \end{align*} By the [Monotone Convergence Theorem](/theorems/626), $\{a_n\}$ converges. The limit is Euler's number $e \approx 2.71828$. [/example] ### Bolzano-Weierstrass and Subsequences A subsequence of $\{a_n\}$ is a sequence $\{a_{n_k}\}_{k=1}^\infty$ where $n_1 < n_2 < n_3 < \cdots$ is a strictly increasing sequence of indices. Subsequences can converge even when the original sequence diverges: the sequence $(-1)^n$ diverges, but the subsequences $\{a_{2k}\} = \{1, 1, 1, \ldots\}$ and $\{a_{2k+1}\} = \{-1, -1, -1, \ldots\}$ both converge. The Bolzano-Weierstrass theorem guarantees that bounded sequences always have *some* convergent subsequence: [quotetheorem:628] The proof by bisection is constructive: at each step, choose the half-interval containing infinitely many terms and extract one. The nested intervals shrink to a point (by the [Cantor Intersection Theorem](/theorems/624)), and the extracted terms converge to that point. This result is equivalent to the statement that [closed, bounded subsets of $\mathbb{R}^n$ are compact](/theorems/315) (the Heine-Borel theorem) — sequential compactness and covering compactness coincide in $\mathbb{R}^n$. Bolzano-Weierstrass fails in infinite-dimensional normed spaces: the standard basis vectors $e_1, e_2, \ldots$ in $\ell^2$ satisfy $\|e_n\| = 1$ (bounded) but $\|e_m - e_n\| = \sqrt{2}$ for $m \neq n$ (no convergent subsequence). This failure is one reason that weak topologies and weak compactness are essential in functional analysis. ## Limits in Metric and Topological Spaces The $\varepsilon$-$N$ definition generalises immediately from $\mathbb{R}$ to any metric space by replacing $|a_n - L|$ with $d(x_n, x)$. In topological spaces where no distance exists, convergence is defined using open sets. ### Metric Space Convergence In a metric space $(X, d)$, a sequence $\{x_n\}$ converges to $x \in X$ if $d(x_n, x) \to 0$ — that is, for every $\varepsilon > 0$ there exists $N$ with $d(x_n, x) < \varepsilon$ for all $n \ge N$. The algebraic rules for sums and products do not apply in a general metric space (there may be no algebraic operations), but the fundamental properties persist: convergent sequences are bounded, limits are unique, and convergent sequences are Cauchy. [definition:Convergence In A Metric Space] Let $(X, d)$ be a metric space. A sequence $\{x_n\}_{n=1}^\infty$ in $X$ **converges** to $x \in X$ if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ with \begin{align*} n \ge N \implies d(x_n, x) < \varepsilon. \end{align*} We write $x_n \to x$ or $\lim_{n \to \infty} x_n = x$. [/definition] [example:Convergence In Function Spaces] In the space $C([0,1])$ of continuous functions on $[0,1]$ with the supremum metric $d_\infty(f, g) = \sup_{x \in [0,1]} |f(x) - g(x)|$, convergence $f_n \to f$ means **[uniform convergence](/page/Uniform%20Convergence)**: $\sup_{x} |f_n(x) - f(x)| \to 0$. With the $L^2$ metric $d_2(f, g) = (\int_0^1 |f(x) - g(x)|^2 \, d\mathcal{L}^1)^{1/2}$, convergence means **convergence in $L^2$-norm**: $\int |f_n - f|^2 \to 0$. The same set $C([0,1])$ carries different notions of convergence depending on the metric, and a sequence can converge in one metric but not the other. [/example] ### Topological Convergence and Nets In a general [topological space](/page/Topology), convergence is defined without a distance function: a sequence $\{x_n\}$ converges to $x$ if for every [open set](/page/Open%20Set) $U$ containing $x$, there exists $N$ with $x_n \in U$ for all $n \ge N$. In metric spaces this coincides with $d(x_n, x) \to 0$ (since the open balls form a neighbourhood basis). In non-first-countable topological spaces, sequences do not suffice to determine the topology. The correct generalisation is the **net** — a function from a directed set (not necessarily $\mathbb{N}$) into the space. A net $\{x_\alpha\}_{\alpha \in A}$ converges to $x$ if for every open $U \ni x$, there exists $\alpha_0 \in A$ with $x_\alpha \in U$ for all $\alpha \ge \alpha_0$. Every result about sequential limits in metric spaces has a net-theoretic analogue in general topological spaces: a set is [closed](/page/Closed%20Set) if and only if it is closed under net limits, a function is continuous if and only if it preserves net convergence, and so on. ## Limits of Functions The limit of a function $f(x)$ as $x \to a$ captures the behaviour of $f$ near $a$ without using the value $f(a)$ itself — indeed, $f$ need not even be defined at $a$. This concept is the foundation of the derivative ($\lim_{h \to 0} (f(a+h) - f(a))/h$) and the definitive test for [continuity](/page/Continuity) ($f$ is continuous at $a$ if and only if $\lim_{x \to a} f(x) = f(a)$). ### The $\varepsilon$-$\delta$ Definition The $\varepsilon$-$\delta$ definition parallels the $\varepsilon$-$N$ definition for sequences, with $\delta$ (a distance threshold) replacing $N$ (an index threshold). [definition:Limit Of A Function] Let $f: D \to \mathbb{R}$ where $D \subseteq \mathbb{R}$, and let $a$ be a limit point of $D$ (i.e., every open interval around $a$ contains a point of $D$ other than $a$). We say $\lim_{x \to a} f(x) = L$ if for every $\varepsilon > 0$ there exists $\delta > 0$ such that \begin{align*} x \in D, \quad 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon. \end{align*} [/definition] The condition $0 < |x - a|$ excludes $x = a$ itself — the limit depends only on the behaviour of $f$ *near* $a$, not *at* $a$. This is essential for the definition of the derivative, where $f$ is evaluated at $a + h$ with $h \neq 0$. ### Sequential Characterisation The $\varepsilon$-$\delta$ definition and the sequential definition are equivalent: $\lim_{x \to a} f(x) = L$ if and only if for every sequence $\{x_n\}$ in $D \setminus \{a\}$ with $x_n \to a$, we have $f(x_n) \to L$. This equivalence — proved by contrapositive in both directions — means that sequence arguments can always replace $\varepsilon$-$\delta$ arguments for function limits. The sequential characterisation is often easier to use: to show $\lim_{x \to 0} \sin(1/x)$ does not exist, exhibit two sequences $x_n = 1/(2\pi n) \to 0$ and $y_n = 1/(2\pi n + \pi/2) \to 0$ with $\sin(1/x_n) = 0$ and $\sin(1/y_n) = 1$ — the function limits along different sequences disagree, so the overall limit does not exist. ### One-Sided Limits and Limits at Infinity The definition extends naturally to one-sided limits ($x \to a^+$ or $x \to a^-$) and to limits at infinity ($x \to \infty$ or $x \to -\infty$). [definition:One Sided Limit] Let $f: D \to \mathbb{R}$ with $a$ a limit point of $D \cap (a, \infty)$. The **right-hand limit** $\lim_{x \to a^+} f(x) = L$ means: for every $\varepsilon > 0$ there exists $\delta > 0$ with $x \in D,\; 0 < x - a < \delta \implies |f(x) - L| < \varepsilon$. The **left-hand limit** $\lim_{x \to a^-} f(x)$ is defined symmetrically. [/definition] The full limit $\lim_{x \to a} f(x) = L$ exists if and only if both one-sided limits exist and agree: $\lim_{x \to a^+} f(x) = L = \lim_{x \to a^-} f(x)$. This criterion is the standard tool for detecting discontinuities at a point: the function $f(x) = \operatorname{sgn}(x)$ has $\lim_{x \to 0^+} f(x) = 1$ and $\lim_{x \to 0^-} f(x) = -1$, so $\lim_{x \to 0} f(x)$ does not exist. [example:Limit Of A Rational Expression] We compute $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$. The function is undefined at $x = 2$ (division by zero), but for $x \neq 2$: \begin{align*} \frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2} = x + 2. \end{align*} Since $f(x) = x + 2$ for $x \neq 2$, and $\lim_{x \to 2}(x + 2) = 4$, we have $\lim_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$. The limit depends only on the behaviour near $x = 2$, not at $x = 2$. [/example] ## Cauchy Sequences and Completeness The definitions above all require knowing the limit $L$ in advance. But in many situations — constructing $\sqrt{2}$, defining the exponential function, solving differential equations — the limit is the unknown. A different convergence criterion is needed: one that detects convergence without reference to the limit. This is the Cauchy criterion. ### Cauchy Sequences A sequence is Cauchy if its terms become arbitrarily close to *each other* (not to any fixed point) as the indices grow. [definition:Cauchy Sequence] A sequence $\{x_n\}_{n=1}^\infty$ in a metric space $(X, d)$ is a **Cauchy sequence** if for every $\varepsilon > 0$ there exists $N \in \mathbb{N}$ such that \begin{align*} m, n \ge N \implies d(x_m, x_n) < \varepsilon. \end{align*} [/definition] Every convergent sequence is Cauchy: if $x_n \to x$, then $d(x_m, x_n) \le d(x_m, x) + d(x, x_n) < \varepsilon/2 + \varepsilon/2$ for large $m, n$. The converse — every Cauchy sequence converges — is not automatic and is the defining property of complete spaces. ### Completeness A metric space where every Cauchy sequence converges is **complete**. Completeness is the property that ensures "limits of approximations exist" — without it, the constructions of analysis break down. [definition:Complete Metric Space] A metric space $(X, d)$ is **complete** if every Cauchy sequence in $X$ converges to a point in $X$. [/definition] The real numbers $\mathbb{R}$ are complete (this is equivalent to the least upper bound property), and $\mathbb{R}^n$ with any of the standard metrics is complete. The rational numbers $\mathbb{Q}$ are not complete: the sequence $1, 1.4, 1.41, 1.414, \ldots$ (decimal approximations to $\sqrt{2}$) is Cauchy in $\mathbb{Q}$ but has no limit in $\mathbb{Q}$. The passage from $\mathbb{Q}$ to $\mathbb{R}$ is precisely the *completion* — filling in the "gaps" by adding limits of Cauchy sequences. In analysis, the most important complete metric spaces are the [Banach spaces](/page/Banach%20Space) (complete [normed vector spaces](/page/Normed%20Vector%20Space)): $L^p$ spaces, [Sobolev spaces](/page/Sobolev%20Space), the space of bounded linear operators, and the space of continuous functions with the supremum norm. The completeness of these spaces is what makes existence theorems (Banach fixed point, Lax-Milgram, spectral theory) work. [example:Completeness Of C With Supremum Norm] The space $C([0,1])$ with the supremum norm $\|f\|_\infty = \sup_{x \in [0,1]} |f(x)|$ is complete. If $\{f_n\}$ is Cauchy, then for each $x$, $\{f_n(x)\}$ is Cauchy in $\mathbb{R}$ (since $|f_n(x) - f_m(x)| \le \|f_n - f_m\|_\infty$), so it converges to some $f(x)$. The Cauchy condition gives $\|f_n - f_m\|_\infty < \varepsilon$ for $m, n \ge N$; letting $m \to \infty$, $\|f_n - f\|_\infty \le \varepsilon$, so $f_n \to f$ uniformly. The uniform limit of continuous functions is continuous, so $f \in C([0,1])$. [/example] ## Interchange of Limits One of the most subtle aspects of the limit concept is the question: when does the order of two limiting operations matter? In general, $\lim_{n \to \infty} \lim_{m \to \infty} a_{n,m} \neq \lim_{m \to \infty} \lim_{n \to \infty} a_{n,m}$, and the failure of interchange is a source of many errors and counterexamples in analysis. ### The Problem Consider a double sequence $a_{n,m}$ or a sequence of functions $f_n(x)$. The two natural limits — first $n$ then $m$, or first $m$ then $n$ — need not agree. [example:Failure Of Limit Interchange] Define $a_{n,m} = n/(n+m)$. Then: \begin{align*} \lim_{n \to \infty} \left(\lim_{m \to \infty} \frac{n}{n+m}\right) = \lim_{n \to \infty} 0 = 0, \qquad \lim_{m \to \infty} \left(\lim_{n \to \infty} \frac{n}{n+m}\right) = \lim_{m \to \infty} 1 = 1. \end{align*} The two iterated limits exist but are unequal. In function terms: $f_n(x) = nx/(n + x)$ on $[0, \infty)$ has pointwise limit $\lim_{n \to \infty} f_n(x) = x$, but $\lim_{x \to \infty} f_n(x) = n$, and the two operations do not commute. [/example] ### Uniform Convergence as the Remedy The standard condition ensuring that limits commute is **uniform convergence**: if $f_n \to f$ uniformly and each $f_n$ is continuous at $a$, then $f$ is continuous at $a$ and $\lim_{x \to a} \lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} \lim_{x \to a} f_n(x)$. The [Cauchy Criterion for Uniform Convergence](/theorems/257) provides a way to verify uniform convergence without knowing the limit function. More generally, uniform convergence allows the interchange of limits with integration ($\lim \int f_n = \int \lim f_n$ under uniform convergence on a bounded domain), with [differentiation](/page/Derivative) (under additional hypotheses — uniform convergence of derivatives), and with summation (the Weierstrass $M$-test). The failure of pointwise convergence to permit these interchanges — the reason that pointwise limits of continuous functions need not be continuous, [integrable](/page/Integral), or differentiable — is one of the central difficulties that motivates the Lebesgue theory of integration and the development of Sobolev spaces. ## References - Rudin, W., *Principles of Mathematical Analysis* (3rd ed., 1976). - Abbott, S., *Understanding Analysis* (2nd ed., 2015). - Munkres, J. R., *Topology* (2nd ed., 2000). - Bartle, R. G. and Sherbert, D. R., *Introduction to Real Analysis* (4th ed., 2011).