Let $X$ be a set, $(Y, d)$ be a complete metric space, and $(f_n)_{n\in \mathbb{N}}$ a sequence of functions satisfying the Cauchy condition $\forall \epsilon > 0:\exists n_0 \in \mathbb{N}:d(f_n(x), f_m(x)) < \epsilon, \forall n, m \geq n_0, x \in X$. Then does it hold true that, similarly, for any $\epsilon > 0$ there exists a large enough $N \in \mathbb{N}$ that $\sup_{x \in X}d(f_n(x), f_m(x)) < \epsilon, \forall n, m \geq N$? We certainly can force the distance to be less than epsilon at any singe point, and therefore at any finite subset of $X$. But I'm not sure how you could proceed with the proof.
The context of this question is, is that a given hint for the problem of showing that a function sequence being Cauchy in a complete metric space is equivalent for that sequence to converge uniformly, was to consider $\sup_{x \in X}d(f_n(x), f_m(x))$.