# Proof of theorem (Limit)

## Homework Statement

Prove that if f(x)<=g(x) then lim f(x) <= lim g(x).

-

## The Attempt at a Solution

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

phinds
Gold Member
2021 Award
Does the limit of a function ever give you a value that is higher than the range of values of the function itself?

HallsofIvy
Homework Helper
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.)

phinds
Gold Member
2021 Award
I am puzzled by phind's question. The answer is "yes, it does" but I don't see how that helps here.

QUOTE]

Hm ... I guess I'm missing something. I don't see how the limit of a function could possibly give you a value that is outside the range of the possible values of the function.

LCKurtz
Homework Helper
Gold Member
Hm ... I guess I'm missing something. I don't see how the limit of a function could possibly give you a value that is outside the range of the possible values of the function.

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

Last edited:
phinds