Need to show that a limit exists

[Demon117]

Homework Statement

Suppose that $$f_{n}$$:R$$\rightarrow$$R convergy uniformly to f. If each function $$f_{n}$$ satisfies $$f_{n}$$$$\rightarrow$$ 0 as x$$\rightarrow$$$$\infty$$, prove that f$$\rightarrow$$0 as x$$\rightarrow$$$$\infty$$. That is show that the limit exists.

Homework Equations

Definition of uniform convergense
Uniform convergence implies pointwise convergence
definition of limits at infinity

The Attempt at a Solution

I have tried to make an estimation using the definition of limits at infinity. I have no idea what I am doing incorrectly but I keep getting that |f(x)|<0 for some reason.

Any advice on this one?

 

[Demon117]

Homework Statement

Suppose that $$f_{n}$$:R$$\rightarrow$$R convergy uniformly to f. If each function $$f_{n}$$ satisfies $$f_{n}$$$$\rightarrow$$ 0 as x$$\rightarrow$$$$\infty$$, prove that f$$\rightarrow$$0 as x$$\rightarrow$$$$\infty$$. That is show that the limit exists.

Also, why is my latex so screwed up?