Proof to show Sequence is Cauchy correct?

Question

Given the two facts

1. The sequence $\{ x_n \}$ is Cauchy in the $L^2(D)$ norm.
2. For each $x$ we can apply the following inequality $\|x\|_{E(D)} \leq k \|x\|_{L^2(D)}$ where $k > 0$

I want to show that $\{ x_n \}$ is Cauchy in the $E(D)$ norm as well. Here is my attempt.

Since $\{ x_n \}$ is Cauchy in the $L^2(D)$ norm, there exists a positive integer $N$ such that for all natural numbers $m,n >N$ we have

$\|x_m -x_n\|_{L^2(D)}< \varepsilon$

Then using fact (2) we get

$\|x_m -x_n\|_{E(D)}< k \|x_m -x_n\|_{L^2(D)}< k \varepsilon$

Take $\varepsilon_{new} = k \varepsilon$ then this completes the proof that $\{ x_n \}$ is Cauchy in the $E(D)$ norm as well.

Does this work?

Answer 1

Proof is correct. In fact, you have proven: if $T:X\to Y$ is linear and continuous and $(x_n)$ is a Cauchy sequence in $C$ then $(Tx_n)$ is a Cauchy sequence in $Y$.