Uniformly convergent sequence proof

[function22]

Homework Statement

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].

Homework Equations

The definition of uniform convergence.

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?

 

[jbunniii]

Homework Statement

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].

Homework Equations

The definition of uniform convergence.

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?

This seems like a promising first step:

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