Does the Space of Functions have a Compact Covering?

Let $X$ be an arbitrary set and $\mathbb{R}^X$ denote the set of real valued functions on $X$. Define the metric $$d(f,g) = \sum_{i=1}^\infty \frac{i \wedge \sup_{x \in X} |f(x) - g(x)|}{2^i}$$ on $\mathbb{R}^X$. Is it true that $\mathbb{R}^X$ has a compact covering? i.e. a collection of compact subsets $K_n$ such that $\cup_{n=1}^\infty K_n = \mathbb{R}^X$?

I feel like the answer should be no, since for the real numbers one uses sets analogous to $\{f \in \mathbb{R}^X: d(f,0) \leq n\}$ and I cannot verify compactness of these sets. Any help would be massively appreciated!