Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 2F (Topics in Analysis)

The closed-set version of the Baire category theorem says that if a nonempty complete metric space is a countable union of closed sets, at least one of those closed sets has nonempty interior. Equivalently, a nonempty complete metric space cannot be a countable union of closed nowhere dense sets.

Fix $ϵ>0$. For each $N$, the set $Q_N={\bigcap }_{n,m\ge N}\{x:|f_n(x)-f_m(x)|\le ϵ\}$ is closed, because each $f_n-f_m$ is continuous. At every $x$, pointwise convergence makes $(f_n(x))$ a Cauchy sequence, so $\mathbb{R}={\bigcup }_{N\ge 1}{Q}_N$. The Baire category theorem supplies an $N_0$ for which $Q_{N_0}$ contains a nonempty open interval $I$. For $x\in I$ and $n\ge N_0$, pass $m\to \infty$ in $|f_n(x)-f_m(x)|\le ϵ$ to obtainThis local uniform Cauchy control for pointwise convergent continuous functions allows both $I$ and $N_0$ to depend on $ϵ$.