Need to show that a limit exists

[Demon117]

Homework Statement

Suppose that f_{n}:R\rightarrowR convergy uniformly to f. If each function f_{n} satisfies f_{n}\rightarrow 0 as x\rightarrow\infty, prove that f\rightarrow0 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\rightarrowR convergy uniformly to f. If each function f_{n} satisfies f_{n}\rightarrow 0 as x\rightarrow\infty, prove that f\rightarrow0 as x\rightarrow\infty. That is show that the limit exists.

Also, why is my latex so screwed up?