Define Fn(x)=exp(-nx) for n=1,2,... . Show that Fn converges uniformly on (1, 00).
Claim: $\displaystyle f_n(x)$ converges uniformly to $\displaystyle f(x) = 0$.
Proof: Let $\displaystyle \epsilon > 0$ be given. Choose $\displaystyle N \in \mathbb{N}$ so that $\displaystyle N > \ln \frac 1 \epsilon$.
Then for all $\displaystyle x \in (1, \infty)$ and $\displaystyle n>N$ we have $\displaystyle |f_n(x) - f(x)| = |e^{-nx}| \le |e^{-n}| < |e^{-N}| < |e^{\ln \epsilon}| = \epsilon$
QED