A set, which appropriately scaled is expressible as sums of elements of a compact set is pre-compact

Question

Assume $X$ is a Banach space and $K\subseteq X$ is compact. Let $C\subseteq X$ be such that $(\forall x\in C)(\exists x_1,x_2\in K)(2x=x_1+x_2)$

Does it follow that $C$ is pre-compact? In particular I am trying to prove this result.

Answer 1

Yes, it does follow. Addition and scalar multiplication are continuous, thus $C$ is a subset of the image of the compact set $K \times K$ under the continuous map $(x,y) \mapsto \frac12(x+y)$.

A subset of a compact set is precompact.