Proving the Non-Negativity of a Limit: Does g(x) > 0 Imply M > 0?

[bjgawp]

Homework Statement

I've got another question involving epsilon-delta proofs, one that is less concrete:
Prove that if g(x) \geq 0 near c and \lim_{x \to c} g(x) = M then M \geq 0. Furthermore, if g(x) > 0 does it follow that M > 0?

Homework Equations

The Attempt at a Solution

Starting off with some preliminary work:
Let \epsilon > 0. We must find \delta > 0 such that |g(x) - M| < \epsilon whenever 0 < |x - c| < \delta

Does anyone have a hint they could provide? I'm not even sure how the end result of this proof is suppose to look like so that's a major set-back in proving this. Simpler, concrete examples require finding a \delta in terms of \epsilon but I don't know how that would apply here.

 

[ZioX]

I would do a proof by contradiction.

The furthermore part is true since g is strictly positive on a neighbourhood around c. Although, what can you say about the case where \lim_{x\to+\infty}g(x)=M when g(x)>0 for all x. Is M>0? If you don't know how to evaluate these kinds of limits just think of a proof/disproof informally.

 

[bjgawp]

Hmm following your advice, the most I can come up with is trying to analyze the following
M - \epsilon < g(x) < M + \epsilon
which isn't a lot as the epsilon poses a problem in coming to a contradiction (such as M > g(x) or M > 0). Anything I'm missing ?

 

[ZioX]

Suppose it was negative. Then we know by our delta/epsilon definition of convergence that g(x) must be negative as x gets gets close to c.

Prove this formally.

 

[bjgawp]

Hmm, sorry for my lack of knowledge but I haven't gone through anything involving convergence (Cauchy sequences?). We've just done a few lectures about epsilon-delta proofs but it's hard to apply what we learn from simple,concrete examples to the general cases. In any case, with the suggestion of doing a proof of contradiction, this is what I've gotten:
|g(x)-M|&lt;\epsilon
g(x)+|M|&lt;\epsilon (since L < 0 and f(x) \geq 0)
0&lt;|M|&lt;\epsilon-g(x)
g(x)-\epsilon&lt;M&lt;0&lt;\epsilon-g(x)

At this point, I can see that e - g(x) can certainly be less than zero if we restrict epsilon small enough. Does this seem right? I know I need to solidify this "proof", if you can call it,that even more.

 

[Dick]

Assume M<0. How about choosing epsilon=|M/2|? if |g(x)-M|<|M/2| then since we've assumed M negative, this forces g(x) to be negative.

 

[ZioX]

I made a mistake, the furthermore part is not true. I was picturing continuous functions in my head for some reason. Let that be a hint.