Exercise 9, § 1

Let X be a Baire space, Y a metric space and $(f_n)$ a sequence of continuous mappings of X into Y which converges pointwise to a function $f$. A point $x \in X$ is said to be a point of uniform convergence of the sequence $(f_n)$ if, for each $\varepsilon > 0$, there exists a neighbourhood V of $x$ and an integer $p$ such that $d(f_m(y), f(g_n)) \leq \varepsilon$ for all $y \in V$ and all integers $m \geq p, n \geq p$, $d$ being the metric on Y. Show that the complement S of the set of points of uniform convergence of the sequence $(f_n)$ is meagre in X [cf. Chapter IX, § 5, Exercise 22 a).] Give an example where S is dense in X. [Take $X = Y = \mathbf{R}$, let $n \to r_n$ be a bijection of $\mathbf{N}$ onto $\mathbf{Q}$, and let $(g_n)$ be a sequence of continuous mappings of $\mathbf{R}$ into $[0, 1]$ which converges to the zero function for $x \neq 0$ and to 1 at $x = 0$, the convergence being uniform in every open set of $\mathbf{R}$ which does not contain 0. Then consider the sequence of functions $f_n(x) = \sum_{p=0}^\infty \alpha_p g_n(x - r_p)$, where the sequence $(\alpha_p)$ tends to 0 in a suitable way.]