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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.

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    S/He is asking for a proof without the use of Cauchy sequences.2012-11-11
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    I've said that just a little hint would be fine, no need for a complete answer. Furthermore, it has nothing to do with homework right now. Also, the old question would probably never be seen again, except maybe through search queries. It did have an answer mentioning Cauchy sequences (which is a good answer to the question, but I'm curious about other answers).2012-11-11
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    I guess you can show that $\liminf_{x \to x_0}f(x)=\limsup_{x \to x_0}f(x)$.2012-11-11
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    I believe any proof will use the completeness of the reals.2012-11-11
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    How do you show limits of an infimum and a supremum? This is pretty new to me.2012-11-11
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    @Py42: What do you mean by "a direct way"? In other words: What are the rules of the game? Are we allowed to use inf and sup, the convergence of bounded monotone sequences? Or do you expect a proof in terms of Dedekind cuts?2012-11-11
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    @ChristianBlatter I was thinking of using as much of the delta-epsilon limit definition as possible, and also inequalities that involve delta & epsilon; Nevertheless I wouldn't mind listening to any sort of proof. I guess this problem really does require things beyond what I've mentioned above. edit: To put it in another way, I was curious to see the correlation between what was given and what was to be proved.2012-11-11

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