# Math Help - Uniform Convergence

1. ## Uniform Convergence

Define Fn(x)=exp(-nx) for n=1,2,... . Show that Fn converges uniformly on (1, 00).

2. Originally Posted by Cairo
Define Fn(x)=exp(-nx) for n=1,2,... . Show that Fn converges uniformly on (1, 00).
Claim: $f_n(x)$ converges uniformly to $f(x) = 0$.

Proof: Let $\epsilon > 0$ be given. Choose $N \in \mathbb{N}$ so that $N > \ln \frac 1 \epsilon$.

Then for all $x \in (1, \infty)$ and $n>N$ we have $|f_n(x) - f(x)| = |e^{-nx}| \le |e^{-n}| < |e^{-N}| < |e^{\ln \epsilon}| = \epsilon$

QED