[proofplan] Since $(\mathcal{C}_p, \chi_{\mathrm{pr}})$ is metrisable, compactness reduces to sequential compactness. Given a sequence in $B(r)$, choose $1$-Lipschitz tree-reduced representatives — they form an equicontinuous, uniformly bounded family, so [Arzelà-Ascoli](/theorems/???) extracts a uniformly convergent subsequence with limit $x$ of $1$-variation $\leq r$. Interpolating between the $1$-variation and uniform norms upgrades convergence to $q$-variation for some $q \in (p, 2)$, which is the natural topology making each signature level $S_m$ continuous. Continuity of $S_m$ on $C_q$ then gives convergence in the product topology, i.e. in $\chi_{\mathrm{pr}}$. [/proofplan] [step:Reduce compactness of $B(r)$ to sequential compactness] By the [Hausdorff, separability, and metrisability theorem](/theorems/2504), $(\mathcal{C}_p, \chi_{\mathrm{pr}})$ is metrisable. The subspace $B(r) \subset \mathcal{C}_p$ inherits a metric structure from any metric inducing $\chi_{\mathrm{pr}}$. In a metrisable space, a subspace is compact if and only if it is sequentially compact. We will therefore prove: every sequence $([x_n])_{n=1}^\infty$ in $B(r)$ has a subsequence that converges in $\chi_{\mathrm{pr}}$ to some $[x] \in B(r)$. [guided] The target topology $\chi_{\mathrm{pr}}$ is abstract — it is defined as an initial topology induced by infinitely many signature-level projections — and verifying compactness directly from the open-cover definition would require us to manipulate arbitrary $\chi_{\mathrm{pr}}$-open covers, which is awkward. **The reduction.** In *general* topological spaces, the equivalence between compactness (every open cover has a finite subcover) and sequential compactness (every sequence has a convergent subsequence) fails: there exist compact spaces that are not sequentially compact, and sequentially compact spaces that are not compact. The equivalence is restored under additional axioms; one of them — the cleanest in our setting — is metrisability. **Why metrisability rescues us.** In a metric space (or any first-countable Hausdorff space), the following are equivalent for a subset $K$: \begin{align*} K \text{ compact} \iff K \text{ sequentially compact} \iff K \text{ closed and totally bounded}. \end{align*} This is the content of the metric-space compactness theorem. We use only the first equivalence. **Apply it to our situation.** The [Hausdorff, separability, and metrisability theorem](/theorems/2504) shows that $(\mathcal{C}_p, \chi_{\mathrm{pr}})$ is metrisable: there exists a metric $d_{\mathrm{pr}}$ on $\mathcal{C}_p$ inducing $\chi_{\mathrm{pr}}$. Any subspace $B(r) \subseteq \mathcal{C}_p$ inherits this metric by restriction, and the subspace topology induced by $d_{\mathrm{pr}}\big|_{B(r) \times B(r)}$ coincides with the subspace topology induced by $\chi_{\mathrm{pr}}$. Hence $B(r)$ with its subspace topology is metrisable, and the metric-space compactness equivalence applies: \begin{align*} B(r) \text{ compact in } (\mathcal{C}_p, \chi_{\mathrm{pr}}) \iff B(r) \text{ sequentially compact in } (\mathcal{C}_p, \chi_{\mathrm{pr}}). \end{align*} **What we have to prove.** It now suffices to verify the sequential-compactness side: every sequence $([x_n])_{n \geq 1}$ in $B(r)$ has a subsequence convergent in $\chi_{\mathrm{pr}}$ to a limit $[x] \in B(r)$. Steps 2-5 produce exactly such a subsequence by working with concrete representatives, applying Arzelà-Ascoli on the path side, upgrading to $q$-variation convergence, and transferring back via continuity of the signature levels. [/guided] [/step] [step:Choose tree-reduced representatives with a uniform Lipschitz bound] Fix a sequence $([x_n])_{n=1}^\infty$ in $B(r)$. By definition of $B(r)$, each class admits a tree-reduced representative $x_n^* : [a, b] \to V$ with $1$-variation \begin{align*} \|x_n^*\|_1 \leq r. \end{align*} We may further assume — by reparametrising linearly — that each $x_n^*$ is parametrised proportionally to arc length on $[a, b]$, so that for $a \leq s \leq t \leq b$, \begin{align*} |x_n^*(t) - x_n^*(s)| \leq \|x_n^*\|_{1; [s,t]} \leq \frac{r}{b - a}(t - s). \end{align*} This shows $x_n^*$ is Lipschitz on $[a,b]$ with Lipschitz constant $L := r/(b - a)$, *uniformly in $n$*. Moreover, by translating each $x_n^*$ if needed (which does not change the equivalence class as elements of $\mathcal{C}_p$), we may assume $x_n^*(a) = 0$ for every $n$. Then $|x_n^*(t)| \leq L(b - a) = r$ for every $t$ and $n$. [guided] The first technical move: every class $[x_n]$ in $B(r)$ has many representatives — anything tree-equivalent to a tree-reduced one. We choose representatives that are well-behaved in two ways. *Lipschitz constant.* For paths of bounded $1$-variation, reparametrising by arc length (or proportional to arc length) produces a Lipschitz parametrisation: if $\|x\|_1 = \ell$, the arc-length parametrisation has Lipschitz constant $1$, and rescaling to a common interval $[a, b]$ gives Lipschitz constant $\ell/(b-a) \leq r/(b-a)$. The key point is the *uniform* bound — every $x_n^*$ has Lipschitz constant at most $L = r/(b-a)$, which does not depend on $n$. *Common starting point.* Tree-reduced equivalence is invariant under translations of the path (translating by a constant yields a tree-equivalent path). So we further translate to make $x_n^*(a) = 0$. With Lipschitz constant $L$ and starting at $0$ on an interval of length $b - a$, the values are bounded: \begin{align*} |x_n^*(t)| = |x_n^*(t) - x_n^*(a)| \leq L(t - a) \leq L(b - a) = r. \end{align*} So the family $(x_n^*)$ is *uniformly bounded* by $r$ and *uniformly Lipschitz* with constant $L$. Both properties are exactly what Arzelà-Ascoli requires. [/guided] [/step] [step:Apply Arzelà-Ascoli to extract a uniformly convergent subsequence] The family $\mathcal{F} := \{x_n^* : n \geq 1\}$ is a family of continuous maps $[a, b] \to V$ such that: - *Uniformly bounded*: $\sup_{n, t} |x_n^*(t)| \leq r$. - *Equicontinuous*: every $x_n^*$ is $L$-Lipschitz, so for every $\varepsilon > 0$, taking $\delta = \varepsilon/L$ gives $|x_n^*(t) - x_n^*(s)| \leq L|t - s| < \varepsilon$ uniformly in $n$ whenever $|t - s| < \delta$. The codomain $V$ is a finite-dimensional real vector space, hence a Banach space, and $[a, b]$ is a compact metric space. The [Arzelà-Ascoli theorem](/theorems/???) for $V$-valued continuous functions on a compact metric space states that an equicontinuous, uniformly bounded family is precompact in $C([a,b]; V)$ with the uniform norm. Hence $(x_n^*)$ has a subsequence — relabel it $(x_n^*)$ — converging uniformly on $[a, b]$ to some continuous limit \begin{align*} x : [a, b] \to V. \end{align*} The limit $x$ inherits the Lipschitz constant: for any $a \leq s \leq t \leq b$, passing to the limit $n \to \infty$ in $|x_n^*(t) - x_n^*(s)| \leq L(t - s)$ gives $|x(t) - x(s)| \leq L(t - s)$, so $x$ is $L$-Lipschitz. By the relation between Lipschitz constants and $1$-variation, \begin{align*} \|x\|_1 \leq L(b - a) = r, \end{align*} so $[x] \in B(r)$. [guided] The classical Arzelà-Ascoli theorem says: a family of continuous functions on a compact metric space, with values in a complete finite-dimensional space, is precompact in $C([a,b]; V)$ with the uniform topology if and only if it is uniformly bounded and equicontinuous. We verify both hypotheses: - *Uniform boundedness* follows from $|x_n^*(t)| \leq r$ for every $n$ and $t$, established in the previous step. - *Equicontinuity* follows from the uniform Lipschitz bound: given $\varepsilon > 0$, every $x_n^*$ satisfies $|x_n^*(t) - x_n^*(s)| < \varepsilon$ whenever $|t - s| < \varepsilon/L$, with $\delta = \varepsilon/L$ independent of $n$. Arzelà-Ascoli then provides a subsequence convergent in the uniform (sup) norm. Pointwise limits of $L$-Lipschitz functions are $L$-Lipschitz: for any $s, t$, the inequality $|x_n^*(t) - x_n^*(s)| \leq L|t-s|$ passes to the limit in $n$ because both sides are continuous functions of the value of $x_n^*$ at finitely many points and uniform convergence implies pointwise convergence. Hence the limit $x$ is $L$-Lipschitz, hence has $1$-variation at most $L(b-a) = r$, hence $[x] \in B(r)$. [/guided] [/step] [step:Upgrade uniform convergence to $q$-variation convergence by interpolation] Fix any $q \in (p, 2)$. We show that the subsequence converges to $x$ in $q$-variation, i.e. in $C_q$ (the space of continuous paths of finite $q$-variation, with the $q$-variation seminorm). For any pair of continuous $L$-Lipschitz paths $f, g : [a, b] \to V$, the $1$-variation of their difference is bounded: \begin{align*} \|f - g\|_1 \leq \|f\|_1 + \|g\|_1 \leq 2r. \end{align*} The general interpolation between $\infty$-norm and $1$-variation reads, for any $q \in (1, \infty)$, \begin{align*} \|f - g\|_q \leq \|f - g\|_1^{1/q} \|f - g\|_\infty^{1 - 1/q}. \end{align*} This is the [interpolation between $\infty$-norm and $1$-variation](/theorems/???): for any partition $a = t_0 < t_1 < \cdots < t_N = b$, \begin{align*} \sum_i |f(t_{i+1}) - g(t_{i+1}) - (f(t_i) - g(t_i))|^q &= \sum_i |f(t_{i+1}) - g(t_{i+1}) - (f(t_i) - g(t_i))|^{q-1} \cdot |\cdot| \\ &\leq \|f - g\|_\infty^{q-1} \cdot \sum_i |\cdot| \cdot 2 \\ &\leq 2 \|f - g\|_\infty^{q-1} \|f - g\|_1. \end{align*} Taking the $q$-th root yields the stated bound (with $2^{1/q}$ a harmless constant; we absorb it into the prefactor and write $\|f - g\|_q \leq C_q \, \|f - g\|_1^{1/q} \|f - g\|_\infty^{1 - 1/q}$ with $C_q = 2^{1/q}$). Applying this to $f = x_n^*$, $g = x_m^*$ — both $L$-Lipschitz, hence with $\|f - g\|_1 \leq 2r$: \begin{align*} \|x_n^* - x_m^*\|_q \leq C_q (2r)^{1/q} \, \|x_n^* - x_m^*\|_\infty^{1 - 1/q}. \end{align*} Since $(x_n^*)$ is Cauchy in the uniform norm (as it converges in that norm), and $1 - 1/q > 0$, the right-hand side tends to $0$ as $n, m \to \infty$. Hence $(x_n^*)$ is Cauchy in $q$-variation. By [completeness of $C_q$](/theorems/???), the sequence converges in $C_q$ to some path; by uniqueness of uniform limits and the embedding $C_q \hookrightarrow C([a,b]; V)$ (via the sup norm), the $C_q$-limit must be $x$. [guided] We have a subsequence $(x_n^*)$ converging *uniformly* to $x$, but the projective topology $\chi_{\mathrm{pr}}$ on $\mathcal{C}_p$ is built from the level-wise signature maps $S_m$, which are continuous on $C_q$ for $q \in (p, 2)$ rather than on the uniform-topology space. We must therefore upgrade uniform convergence to $C_q$-convergence. The classical interpolation inequality is: \begin{align*} \|f\|_q \leq C_q \, \|f\|_1^{1/q} \|f\|_\infty^{1 - 1/q}, \qquad q \in (1, \infty). \end{align*} Conceptually, this says: the $q$-variation of an increment is dominated by a *high power* (close to $1$) of the $1$-variation when $q$ is small, and a *high power* of the uniform norm when $q$ is large. The proof rewrites $|\Delta f|^q = |\Delta f|^{q-1} \cdot |\Delta f|$, bounds the first factor by $\|f\|_\infty^{q-1}$, sums to get $\|f\|_1$, and takes the $q$-th root. Applying this to differences: \begin{align*} \|x_n^* - x_m^*\|_q \leq C_q \, \|x_n^* - x_m^*\|_1^{1/q} \|x_n^* - x_m^*\|_\infty^{1 - 1/q}. \end{align*} The first factor is uniformly bounded — both paths are $L$-Lipschitz, so their difference has $1$-variation at most $2r$ — but it is *not* small. The second factor is small because of uniform convergence. The product is small because the second factor's positive exponent $1 - 1/q > 0$ wins. Hence $(x_n^*)$ is Cauchy in the $q$-variation norm, and we identify the $C_q$-limit as $x$ (the same limit produced by uniform convergence, since the $C_q$ topology is finer than the uniform topology on bounded $1$-variation sets). This is exactly the standard 'interpolation upgrade' technique in rough path analysis: pay a small power of the controlled $1$-variation, gain a positive power of the small uniform error. [/guided] [/step] [step:Conclude convergence in $\chi_{\mathrm{pr}}$ by continuity of each signature level] For every $m \geq 0$, the signature truncation \begin{align*} S_m : C_q \to V^{\otimes m}, \qquad x \mapsto \pi_m(S(x)) \end{align*} is continuous on $C_q$ for $q \in (p, 2) \subset [1, 2)$ — this is the [continuity of signature truncations on $q$-variation paths](/theorems/???). Since $x_n^* \to x$ in $C_q$, continuity gives \begin{align*} S_m(x_n^*) \to S_m(x) \quad \text{in } V^{\otimes m} \text{ for every } m \geq 0. \end{align*} Convergence at every level is exactly convergence in the product topology on $\prod_{m \geq 0} V^{\otimes m}$. Pulling back through the homeomorphism $S : (\mathcal{C}_p, \chi_{\mathrm{pr}}) \to \mathcal{S}_p$ from the [definition of $\chi_{\mathrm{pr}}$](/theorems/???), this is exactly the statement that $[x_n] \to [x]$ in $\chi_{\mathrm{pr}}$. We have produced, from the original sequence, a subsequence converging in $\chi_{\mathrm{pr}}$ to a limit $[x] \in B(r)$. Hence $B(r)$ is sequentially compact, and by the metrisability reduction of the first step, $B(r)$ is compact in $(\mathcal{C}_p, \chi_{\mathrm{pr}})$. [guided] The final step transfers the $C_q$ convergence we have established to the projective topology $\chi_{\mathrm{pr}}$ — the topology in which we are required to prove compactness. The projective topology is generated by the level signature maps $S_m$, and the statement that $S_m : C_q \to V^{\otimes m}$ is continuous for $q \in [1, 2)$ is a foundational result in rough path theory: each level of the signature is a continuous functional in $q$-variation as long as $q$ stays below the threshold $2$. We chose $q \in (p, 2)$ specifically so that this continuity result applies. Convergence in $C_q$ then yields convergence of every $S_m(x_n^*) \to S_m(x)$ in the finite-dimensional space $V^{\otimes m}$. The product topology on $\prod_m V^{\otimes m}$ is, by definition, the topology of coordinatewise convergence: $a^{(n)} \to a$ in the product if and only if $\pi_m(a^{(n)}) \to \pi_m(a)$ for every $m$. So our level-by-level convergence is exactly convergence of $S(x_n^*) \to S(x)$ in the product topology, which by the homeomorphism $S$ between $\mathcal{C}_p$ and $\mathcal{S}_p$ is exactly $[x_n] \to [x]$ in $\chi_{\mathrm{pr}}$. The limit point $[x]$ has $\|x\|_1 \leq r$ from the Lipschitz constant we tracked, so $[x] \in B(r)$. This completes the proof of sequential compactness of $B(r)$, hence of compactness in the metrisable space $(\mathcal{C}_p, \chi_{\mathrm{pr}})$. [/guided] [/step]