# Real Analysis: delta epsilon

• May 6th 2008, 08:07 PM
Vitava61
Real Analysis: delta epsilon
Ok, to be honest, I have no idea where to start for this problem, and I',m not sure to to annotate some of it:

Let f: D-->R and let c be an accumulation point of D. Assume lim f(x) = L, L>0
x->c

Prove using the epsilon- delta argument there exists a deleted neighborhood N*(c,epsilon) so that f(x) >0 for all xEN*(x, epsilon) (intersection) D.

That's all that was given, I have a final exam on Friday, any help would be greatly appreciated.
• May 6th 2008, 08:27 PM
TheEmptySet
Quote:

Originally Posted by Vitava61
Ok, to be honest, I have no idea where to start for this problem, and I',m not sure to to annotate some of it:

Let f: D-->R and let c be an accumulation point of D. Assume lim f(x) = L, L>0
x->c

Prove using the epsilon- delta argument there exists a deleted neighborhood N*(c,epsilon) so that f(x) >0 for all xEN*(x, epsilon) (intersection) D.

That's all that was given, I have a final exam on Friday, any help would be greatly appreciated.

Since the limit exists at c, we can find a delta for every epsilon

$\displaystyle \exists \delta > 0, \ni \mbox{ } |x-c|< \delta, |f(x)-L|< \epsilon$

Let $\displaystyle \epsilon=L$ so we can find a $\displaystyle \delta$
such that when
$\displaystyle |x-c|<\delta$ $\displaystyle |f(x)-L|<\epsilon=L$

Now working on the 2nd we have

$\displaystyle -L < f(x)-L< L \iff 0< f(x) < 2L$

By the above when $\displaystyle x \in (c-\delta,c+\delta)$

$\displaystyle f(x) > 0$

QED