Analysis II | Part IB, 2001

Let {fn}\left\{f_{n}\right\} be a pointwise convergent sequence of real-valued functions on a closed interval [a,b][a, b]. Prove that, if for every x[a,b]x \in[a, b], the sequence {fn(x)}\left\{f_{n}(x)\right\} is monotonic in nn, and if all the functions fn,n=1,2,f_{n}, n=1,2, \ldots, and f=limfnf=\lim f_{n} are continuous, then fnff_{n} \rightarrow f uniformly on [a,b][a, b].

By considering a suitable sequence of functions {fn}\left\{f_{n}\right\} on [0,1)[0,1), show that if the interval is not closed but all other conditions hold, the conclusion of the theorem may fail.

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