Uniform convergence of a series

[WaterPoloGoat]

Homework Statement

We know that f is uniformly continuous.

For each n in N, we define f_n(x)=f(x+1/n) (for all x in R).

Show that f_n converges uniformly to f.

Homework Equations

http://en.wikipedia.org/wiki/Uniform_convergence

The Attempt at a Solution

I know that as n approaches infinity, that f_n(x)=f(x), which implies that f_n converges to f.

I'm currently trying to apply the fact that if f_n is uniformly convergent, then

lim_n->infinity Sup {f_n(x): x in R}=0.

But I keep getting stuck on the fact that there's an function in the definition of f_n i.e., f_n(x)=f(x+1/n). Is there a way to work with it?

 

[losiu99]

It's actually pretty straightforward. Given $$\epsilon > 0$$, for each x there is a $$\delta>0$$ such that $$|f(x+h)-f(x)|<\epsilon$$ whenever $$|h|<\delta$$. Hence, for all $$n > \delta ^{-1}$$ we have $$|f_n(x)-f(x)|<\epsilon$$.