 # 1. For each n € N let fn : [2, xo) R be given by fn (x) = 1+xn...

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1. For each n € N let fn : [2, xo) R be given by fn (x) = 1+xn 1 Find the function f : [2,00) R to which {fn} converges pointwise. Prove that the convergence is uniform. 2. For each n E N let fn : [1, xo) R be given by fn (x) = 1+xn 1 Find the function f : [1,00) R to which {fn} converges pointwise. Prove that the convergence is not uniform. Extra credit: Does {fn} converge uniformly to f on (1,00): Justify your answer. 3. For each n € N let fn : [0,1] R be given by fn (x) = n if x € (0,1/) 0 otherwise. (a) Find the function f : [0,1] R to which {fn} converges pointwise. (b) For each n E N compute so fn. Does {s' fn} converge to forf? (c) Can the convergence of {fn} to f be uniform? 4. For each n € N let fn : R R be given by fn (x) = x² 1 Recall that in class we showed {fn} converges uniformly to the function f : R R given by f (x) = x|. (a) Prove that {fh} converges pointwise to the function g : R R given by 1 if x < 0 g(x) = if x = 0 1 if x > 0. (b) Can the convergence of {f'}} to g be uniform?

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