A pointwise limit of bounded functions need not be bounded. For \(n\geq 1\) let \(f_n\colon(0,1)\to\RR\), \(f_n(x) \coloneqq \frac{1}{x + \frac1n}\). Each \(f_n\) is bounded, since \(0 < f_n(x) \leq n\) for \(x\in(0,1)\). For each \(x\in(0,1)\), \(f_n(x)\to \frac1x\), and \(x\mapsto\frac1x\) is unbounded on \((0,1)\).
Give a sequence of bounded functions whose pointwise limit is unbounded.