Uniformly Convergent Sequence on Dense Subset

Theorem
Let $X$ be a metric space.

Let $Y\subset X$ be dense.

Let $V$ be a Banach space.

Let $(f_n)$ be a sequence of continuous mappings $f_n : X\to V$.

Let $(f_n)$ be uniformly convergent on $Y$.

Then $(f_n)$ is uniformly convergent on $X$.

Proof
Let $\epsilon>0$.

Let $N\in\N$ be such that $\Vert f_n-f_m\Vert < \epsilon$ for $n,m > N$ on $Y$.

Let $x\in X$.

Then there exists a sequence $(y_n)\in Y$ with $y_n\to x$.

By continuity, $\Vert f_n(x)-f_m(x)\Vert \leq \epsilon$.

Because $V$ is complete, $(f_n)$ is uniformly convergent on $X$.