Does the image of a Cauchy sequence converge?

Question

Let $V$ be a normed space and $W$ a Banach space. Given a bounded linear operator $A \in \mathcal L(V,W)$ and a Cauchy sequence $\{f_n\}_{n\in \mathbb N}$ in $V$, does the image sequence, $\{Af_n\}_{n\in \mathbb N}$, converge? I have the following proof: since $\{f_n\}$ is Cauchy, $\forall \varepsilon > 0: \exists N\in \mathbb N: \forall n, m \ge N:\|f_n-f_m\| < \frac\varepsilon{\|A\|}.$ So, we then have $$\|Af_n - Af_m\| =\|A(f_n -f_m)\| \le \|A\|\cdot\|f_n-f_m\|<\varepsilon, $$ so that the sequence $\{Af_n\}_{n\in \mathbb N}$ is Cauchy in $W$. But, $W$ is a Banach space, so the sequence converges. Is my proof correct?