Proving a theorem about the limit of a function The theorem is as follows:
$$\exists L\in\mathbb R\ \bigg(\lim_{x \to c}f(x)=L\bigg)\iff\forall\varepsilon>0\ \exists\delta>0\ \forall x_1,x_2\in B_{\delta}(c):|f(x_1)-f(x_2)|<\varepsilon$$
where $B_{\delta}$ is the set $\{x\in \mathscr D_f \ | \ 0<|x-c|<\delta\}$
It is quite easy to prove the implication $LHS\Rightarrow RHS$.
It is the other one I am struggling with( $LHS\Leftarrow RHS$ ).I'd appreciate a sketch of the proof or a hint on how to approach this problem. Thanks.

David, some ideas for you:
We assume $\;\lim_{x\to c}f(x)=L\;$ (i.e., the limit exists and equals $\;L\;$), and by definition this means that 
$$\forall\,\epsilon>0\;\;\exists\,\delta>0\;\;s.t.\;\;|x-c|<\delta\implies |f(x)-L|<\frac\epsilon2$$
so if $\;x_1\,,\,x_2\;$ are two such values (i.e., $\;|x_i-c|<\delta\,,\,i=1,2\;$) , then we get
$$|f(x_1)-f(x_2)|=|f(x_1)-L+L-f(x_2)|\le|f(x_1)-L|+|f(x_2)-L|<\frac\epsilon2+\frac\epsilon2=\epsilon$$
and we're done.
