Suppose we have a sequence $\{v_j\}_{j\in\mathbb{N}}$ in a Banach space $X$, and suppose that for each $\varepsilon>0$, there is some $N\geq0$ and some bounded sequence $\{x_j\}_{j\in\mathbb{N}}$ in $\mathbb{C}^N$ such that
$$\lVert v_s-v_k\rVert\leq\varepsilon+\lVert x_s-x_k\rVert_\infty.$$
Can we then conclude that $\{v_j\}_{j\in\mathbb{N}}$ has a Cauchy subsequence? I think the answer to this is yes, and the following is my reasoning:
Start with $\varepsilon=1$. As $\{x_j\}_{j\in\mathbb{N}}$ is bounded in $\mathbb{C}^N$, it has a Cauchy subsequence $\{x_{n^1_j}\}_{j\in\mathbb{N}}$. Consequently
$$\limsup_{s,k\to\infty}\lVert v_{n_s^1}-v_{n_k^1}\rVert\leq1.$$
Repeat the argument with $\varepsilon=\frac{1}{2}$, but extract instead a subsequence $\{v_{n^2_j}\}_{j\in\mathbb{N}}$ of $\{v_{n^1_j}\}_{j\in\mathbb{N}}$ such that
$$\limsup_{s,k\to\infty}\lVert v_{n_s^1}-v_{n_k^1}\rVert\leq\frac{1}{2}.$$
Repeat this inductively for all $m\in\mathbb{N}$ to obtain nested subsequences with $\{v_{n_j^m}\}_{j\in\mathbb{N}}$ a subsequence of $\{v_{n_j^{m-1}}\}_{j\in\mathbb{N}}$ with the property that
$$\limsup_{s,k\to\infty}\lVert v_{n_s^m}-v_{n_k^m}\rVert\leq\frac{1}{m}.$$
My hope is now that the diagonal sequence $\{v_{n_j^j}\}_{j\in\mathbb{N}}$ should be the desired Cauchy subsequence, however I am not sure whether this is actually true, and whether what I've done so far even works. My idea from here on is to let $\varepsilon>0$ and find an $M\in\mathbb{N}$ such that
$$\frac{1}{M}<\frac{\varepsilon}{2}.$$
Then I find an $N\in\mathbb{N}$ such that if $s,k\geq N$, then
$$\lVert v_{n_s^M}-v_{n_k^M}\rVert\leq\frac{1}{M}+\frac{\varepsilon}{2}<\varepsilon.$$
In particular, if $s,k\geq\max\{N,M\}$, then
$$\lVert v_{n_s^s}-v_{n_k^k}\rVert<\varepsilon,$$
as we have taken consecutive subsequences, proving that the sequence is indeed Cauchy.
What I'm asking is whether what I claim is even correct, and if so whether my argument for it (or at least the idea behind it) works, and if not if there is a better way of doing it. The origin of this question is that I'm trying to show that a certain linear operator is compact, and if I can show this claim then that will follow.