Prove: Let $\displaystyle X $ be a metric space, let $\displaystyle a \in X $ be a limit point of $\displaystyle X $, and let $\displaystyle f: X \to \mathbb{R} $ be a function. Suppose that $\displaystyle \lim_{x\to a} f(x) = L $. For all $\displaystyle K \in \mathbb{R} $ such that $\displaystyle |L| < K $, there exists $\displaystyle \delta > 0 $ such that if $\displaystyle x \in B_{\delta}(a) \ \backslash \{a \} $ then $\displaystyle |f(x)| < K $

In other words $\displaystyle -K < f(x) < K $ for some deleted $\displaystyle \delta $-ball about $\displaystyle a $. How would you show this? What value of $\displaystyle \delta $ would you pick?