if r S if


if r S<br>if <a<1.<br>Let (fn)n be a sequence of functions fn : [0, 1]→ R defined by fn(r) =<br>nz<br>Then<br>1. f, converges to 0 pointwisely on [0, 1] since fn converges to 0 pointwisely.<br>2. fn converges tof = 0 uniformly on [0, 1] and thus f fn(x)dx converges to ſ f(x)dr.<br>3. fn converges to 0 pointwisely on [0, 1] but not uniformly.<br>4. fn doesn't converge pointwisely to 0 on [0, 1].<br>

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Jun 04, 2022
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