This is how I understand epsilon ($\epsilon$) and delta ($\delta$) relation (please correct me if I'm wrong)-
Let the limit of $f(z) = L$ as $z$ approaches $k$. To prove or see that it is actually the limit,
let us take an $\epsilon$ such that $\epsilon > |f(z) - L |$
If, for every $\epsilon>0$, there is a $\delta>0$ such that $\delta> |z-k|$ , then the limit exists and is equal to $L$.
I'm unable to understand how this works. How does the existence of such a $\delta$ proves that limit is what we assumed?