Prove that $(C[0, T], ρ)$ is a complete metric space.

Question

Let $T > 0$ and $L ≥ 0$. Consider $C[0, T]$, the space of all continuous real valued functions on $[0, T]$, with the metric $ρ$ defined by $ρ(x, y) = sup_{ (0≤t≤T)} e ^{−Lt}|x(t) − y(t)|$.

I want to prove that $(C[0, T], ρ)$ is a complete metric space.I understand that I need to take an arbitrary Cauchy sequence of functions from $C[0,T]$ and prove that it converges to some function in the space. May I please ask how to show that? Thanks so much.

Answer 1

Let $(f_n)$ be your sequence of functions. You need to do three things:

1. Prove that $(f_n(x))$ converges for all $x\in[0,T]$. Call its limit $f(x)$.

2. Prove that $(f_n)$ converges to $f$ uniformly.

3. Deduce that $f$ is continuous.