Proof of theorem (Limit)

1. Oct 5, 2013

cristina89

1. The problem statement, all variables and given/known data
Prove that if f(x)<=g(x) then lim f(x) <= lim g(x).

2. Relevant equations
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3. The attempt at a solution

I've tried by definition of limit, but I didn't get anywhere with this... Can anyone help me??

2. Oct 5, 2013

phinds

Does the limit of a function ever give you a value that is higher than the range of values of the function itself?

3. Oct 5, 2013

HallsofIvy

Staff Emeritus
Try a "proof by contradiction". Suppose lim f(x)> lim g(x). Let $\alpha$= lim f(x)- lim g(x) and choose $\epsilon= \alpha/2$.

I am puzzled by phind's question. The answer is "yes, it does" but I don't see how that helps here.

(Note, by the way, if the condition were "f(x)< g(x)" then it would NOT be true that "lim f(x)< lim g(x)". Phind's suggestion would be helpful in proving that.)

4. Oct 5, 2013

5. Oct 5, 2013

LCKurtz

Try $\lim_{x\to\infty}\frac 1 x$.

Last edited: Oct 5, 2013
6. Oct 5, 2013

phinds

OK, thanks. I got it.