I've asked the question below before with no answer, but I would like to stress that this time it is not a homework question (and also that I've spent hours trying to come up with a solution).
This is the question:
Let f be a function defined around $x_o$. For every $\epsilon>0$ there's some $\delta>0$ such that if $0<|x-x_0|<\delta$ and $0<|y-x_0|<\delta$ then $|f(x)-f(y)|<\epsilon$.
And what's needed to be proven is that $\lim_{x\to x_0}f(x)$ exists.
I've been told that there are two ways to do so: One is quite easy and requires Cauchy sequences (I haven't learned sequences yet, but I think I'll look it up sometime soon and try to solve it this way).
The second way is a direct way, which I've been told is cumbersome and unrecommended, but since this is the way I tried solving it so far, I am really curious as to how the proof goes and this is the way I'm asking about. I tried applying all kinds of inequalities but with no success.
Even a little hint/direction would be swell. Thank you in advance.