conjugations of a neighbourhood of the identity in a locally compact group

Question

Let $G$ be a second countable locally compact group equipped with a left invariant metric $d$ and $V$ be a neighbourhood of the identity $e$. Consider a compact subset $K$ of $G$.

How show that there exists $\varepsilon>0$ such that $tVt^{-1}$ contains the ball $B(e,\varepsilon)$ for any $t \in K$ ?

Answer 1

Define $f : K \to [0, \infty)$ via $$f(t)= \sup \{ \epsilon | B(e,\varepsilon) \subset tVt^{-1} \}$$

Show that $f$ is continuous and not vanishing. By the compactness of $K$, $f$ attains its absolute minimum.