Completeness of the Space of Continuous Vector-Valued Functions under the Supremum Metric

Let $a<b$ be real numbers and let $k\ge1$ be a natural number. Let $\mathcal{C}$ denote the set of all functions $h:[a,b]\to\mathbb{R}^{k}$ (Euclidean space) whose component functions $h^{1},\dots,h^{k}:[a,b]\to\mathbb{R}$ are continuous 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. For $h,h'\in\mathcal{C}$ define

where $d$ is the Euclidean distance on $\mathbb{R}^{k}$ (a metric by Euclidean Distance is a Metric on $\mathbb{R}^n$) and the supremum is taken over the nonempty set of values.

1. The supremum defining $d_{\infty}$ is finite for all $h,h'\in\mathcal{C}$, and $(\mathcal{C},d_{\infty})$ is a nonempty metric space.

2. $(\mathcal{C},d_{\infty})$ is a complete metric space.