# Uniformly convergent sequence proof

1. Feb 26, 2012

### function22

1. The problem statement, all variables and given/known data

Let $$f_n(x)$$ be a sequence of functions that converges uniformly to f(x) on the interval [0, 1]. Show that the sequence $$e^{f_n(x)}$$ also converges uniformly to $$e^{f(x)}$$ on [0,1].

2. Relevant equations

The definition of uniform convergence.

3. The attempt at a solution

I tried to use the definition of uniform convergence to prove this, so I need to show that for all ε>0 there exists N≥1 for all n≥N for all x in [0,1] $$|e^{f_n(x)}-e^{f(x)}| < ε$$. I tried to prove this from the fact that $$f_n(x)$$ converges uniformly to f(x) but I kept getting stuck and I'm not sure how to do this problem now. Can anyone help me please?

2. Feb 27, 2012

### jbunniii

This seems like a promising first step:

$$|e^{f_n(x)} - e^{f(x)}| = e^{f(x)} |e^{f_n(x)-f(x)} - 1|$$