A family of functions can fail to converge for two very different reasons. It may oscillate too much, or it may escape vertically. Uniform boundedness isolates the second obstruction. Before asking whether a sequence of functions has a pointwise limit, a uniformly convergent subsequence, or a convergent integral, we first ask whether all functions in the sequence live inside one fixed vertical strip.

The word "uniformly" matters because ordinary boundedness of each function is too weak. A sequence may consist entirely of bounded functions while the bounds needed for the individual functions grow without limit. In that case there is no single number controlling the whole sequence, and many compactness and convergence arguments lose their first estimate.

[example: Bounded Terms Without a Uniform Bound] Let $E=[0,1]$, and for each $n \in \mathbb N$ define \begin{align*} f_n(x)=nx,\qquad x \in E. \end{align*} For a fixed $n$, every $x \in [0,1]$ satisfies $0 \le x \le 1$. Multiplying this inequality by the positive number $n$ gives \begin{align*} 0 \le nx \le n. \end{align*} Since $f_n(x)=nx$, this is \begin{align*} 0 \le f_n(x) \le n. \end{align*} Thus the single function $f_n$ is bounded on $E$, with bound depending on $n$. The sequence $(f_n)$ is not uniformly bounded on $E$. To see this, let $M \ge 0$ be any proposed common bound. Choose $n \in \mathbb N$ with $n>M$, and evaluate at $x=1 \in E$. Then \begin{align*} f_n(1)=n\cdot 1=n. \end{align*} Hence \begin{align*} |f_n(1)|=|n|=n>M. \end{align*} So this $M$ does not bound all values $|f_n(x)|$ at once. The failure is exactly that the valid bound for the individual term $f_n$ grows with $n$, so there is no single constant controlling the whole sequence. [/example]

This example shows why uniform boundedness is a property of the whole sequence, not a property that can be checked term by term without keeping track of the constants. It is the first layer of control in many arguments: once a common bound exists, pointwise operations, limiting procedures, dominated convergence arguments, and compactness theorems can begin to operate.

## Definition

### Common Bounds

A sequence is an ordered list indexed by $\mathbb N$. In this chapter the terms are functions, so the phrase "bounded sequence" must not be confused with a bounded numerical sequence. The question is whether all the function values, for all indices and all points in the domain, are bounded by the same number.

[definition: Uniformly Bounded Sequence] Let $E \subset \mathbb R$, and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n: E \to \mathbb R$. The sequence $(f_n)$ is uniformly bounded on $E$ if there exists $M \ge 0$ such that \begin{align*} |f_n(x)| \le M \end{align*} for every $n \in \mathbb N$ and every $x \in E$. [/definition]

The order of the quantifiers is the entire definition: one number $M$ must work simultaneously for every function and every point. If $M$ is allowed to depend on $n$, the condition collapses to the separate boundedness of each individual function.

[remark: Quantifier Order] Uniform boundedness says \begin{align*} \exists M \ge 0\,\forall n \in \mathbb N\,\forall x \in E,\quad |f_n(x)| \le M. \end{align*} Separate boundedness of the terms says \begin{align*} \forall n \in \mathbb N\,\exists M_n \ge 0\,\forall x \in E,\quad |f_n(x)| \le M_n. \end{align*} The first statement is stronger because the bound is chosen before the index $n$ is known. [/remark]

### Norm Language

Many arguments need to replace repeated pointwise estimates by a single numerical sequence. Without a canonical number attached to each function, the phrase "one bound works for all functions" remains tied to two variables at once. For a [bounded function](/page/Bounded%20Function), the relevant obstruction is its largest absolute size on the domain, and taking the supremum records exactly the least possible uniform bound for that function.

[definition: Supremum Norm on Bounded Functions] Let $E \subset \mathbb R$ be nonempty, and let $\mathcal B(E,\mathbb R)$ denote the set of bounded functions from $E$ to $\mathbb R$. The supremum norm on bounded functions is the function $\|\cdot\|_{\infty,E}: \mathcal B(E,\mathbb R) \to [0,\infty)$ defined by \begin{align*} \|f\|_{\infty,E}:=\sup_{x \in E} |f(x)|. \end{align*} [/definition]

With this notation, uniform boundedness can be read as a numerical boundedness condition on the sequence $(\|f_n\|_{\infty,E})$. This reformulation is often the most efficient way to use the definition.

[quotetheorem:9522]

The theorem is not just notation. It says that a question about two variables, the index $n$ and the point $x$, can be compressed into one sequence of norms. Many compactness arguments begin by proving a bound in a norm and then invoking this characterisation.

[example: A Uniform Bound From a Supremum Estimate] Let $E=\mathbb R$, and for each $n \in \mathbb N$ define \begin{align*} f_n(x)=\frac{\sin(nx)}{1+x^2},\qquad x \in \mathbb R. \end{align*} Fix $n \in \mathbb N$ and $x \in \mathbb R$. The sine function satisfies $-1 \le \sin(nx) \le 1$, so $|\sin(nx)| \le 1$. Also $x^2 \ge 0$, hence \begin{align*} 1+x^2 \ge 1. \end{align*} In particular $1+x^2>0$, so \begin{align*} |f_n(x)|=\left|\frac{\sin(nx)}{1+x^2}\right|. \end{align*} Since the denominator is positive, \begin{align*} \left|\frac{\sin(nx)}{1+x^2}\right|=\frac{|\sin(nx)|}{1+x^2}. \end{align*} Using $|\sin(nx)| \le 1$ gives \begin{align*} \frac{|\sin(nx)|}{1+x^2} \le \frac{1}{1+x^2}. \end{align*} Using $1+x^2 \ge 1$ gives \begin{align*} \frac{1}{1+x^2} \le 1. \end{align*} Combining these inequalities, \begin{align*} |f_n(x)| \le 1. \end{align*} The same constant $1$ works for every $n \in \mathbb N$ and every $x \in \mathbb R$, so $(f_n)$ is uniformly bounded on $\mathbb R$ with common bound $M=1$. The estimate controls only the vertical size of the functions; it does not prevent the oscillations of $\sin(nx)$ from becoming faster as $n$ grows. [/example]

The example separates vertical control from oscillatory control. Uniform boundedness does not say that nearby points have nearby values, and it does not say that the sequence converges. It says only that the sequence cannot escape to arbitrarily large heights.

## Uniform Versus Pointwise Control

### Pointwise Bounds