Dilation of Compact Set in Topological Vector Space is Compact/Proof 2

Proof
From Dilation Mapping on Topological Vector Space is Continuous, the mapping $c_t : X \to X$ defined by:
 * $\map {c_t} x = t x$

for each $x \in X$ is continuous.

From Continuous Image of Compact Space is Compact:
 * $\map {c_t} K = t K$ is compact.