Theorem regarding not uniformly convergent

Question

Let $f_n$ be a sequence of functions on $A \subseteq \mathbb{R}$ .Show that $f_n$ does not converge uniformly on $A_0 \subseteq A$ to a function $f:A_0 \rightarrow \mathbb{R}$ iff for some $\epsilon_0>0$ there exists a subsequence {$f_{n_k}$} of {$f_n$} and a sequence {$x_k$} in $A_0$ such that $|f_{n_k}(x_k)-f(x_k)| \ge \epsilon_0 \ \ \forall k \in \mathbb{N}$ I saw this as a lemma in Bertle Sherbert, and generally use this trick to disprove certain sequences are uniformly convergent. But I am stuck how to prove this lemma. I believe this is just a negation of the definition of uniform convergence.But I want to add more rigour.