Intersection of conjugates by compact subset of open neighborhood of identity contains open neighborhood of identity

Statement
Suppose $$G$$ is a topological group, $$U$$ is an open subset of $$G$$ containing the identity element, and $$K$$ is a compact subset of $$G$$. Then, there exists an open subset $$V$$ of $$G$$ containing the identity element such that:

$$V \subseteq xUx^{-1} \ \forall \ x \in K$$