Consider fn: [0, π] → R given by fn(x) = sin(x), for n ≥ 1. Prove that the sequence (fn) converges pointwise to f: [0, π] → R given by if x = 0 if x # yet this convergence is not uniform. (See Figure 7.3) 1.0 0.8 0.6 0.4 0.2 0.5 1.0 f(x) = 15 2.0 Figure 7.3: The functions f(x) = sin"(x) 2.5 3.0

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