Arzela-Ascoli Theorem for Vector-Valued Functions on a Compact Interval

Let $a<b$ be real numbers, let $k$ be a natural number, and write $|x|$ for the Euclidean norm on Euclidean space $\mathbb{R}^{k}$. Regard $[a,b]$ as a metric space under the distance $(s,t)\mapsto|s-t|$ and $\mathbb{R}^{k}$ as a metric space under the Euclidean distance.

Let $(f_{n})_{n\in\mathbb{N}}$ be a sequence of maps $[a,b]\to\mathbb{R}^{k}$ such that:

(i) (uniform boundedness) there is a real number $M$ with $|f_{n}(t)|\le M$ for every $n\in\mathbb{N}$ and every $t\in[a,b]$;

(ii) (uniform equicontinuity) the family $\{f_{n}:n\in\mathbb{N}\}$ is uniformly equicontinuous.

Then there exist a subsequence $(f_{n_{j}})_{j\in\mathbb{N}}$ and a map $f:[a,b]\to\mathbb{R}^{k}$, continuous in each component on $[a,b]$, the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric, such that the real sequence

$$\Bigl(\sup_{t\in[a,b]}\bigl|f_{n_{j}}(t)-f(t)\bigr|\Bigr)_{j\in\mathbb{N}}$$

is well defined and has limit $0$.

Moreover the limit inherits the bound and the modulus of the family: $|f(t)|\le M$ for every $t\in[a,b]$, and if $\varepsilon>0$ and $\delta>0$ are as in the uniform equicontinuity of $\{f_{n}:n\in\mathbb{N}\}$, then $|f(s)-f(t)|\le\varepsilon$ for all $s,t\in[a,b]$ with $|s-t|<\delta$.