Problem 6 (6 points) Consider the sequence of continuous functions {fn} on [0, 1] given by...

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Problem 6 (6 points) Consider the sequence of continuous functions {fn} on [0, 1] given by fn(2): = {19 nx 0x <1/n 1/n x 1 Show that {f} has no subsequence which is convergent in uniform norm. (Hint: Show that every subsequence of {f} converges pointwise to some function. Can the subsequences converge uniformly?) (Remark: This is an example of a sequence of continuous functions bounded in the uniform norm which has no convergent subsequence.) Problem 6 (6 points) Consider the sequence of continuous functions {fn} on [0, 1] given by fn(2): = {19 nx 0x <1/n 1/n x 1 Show that {f} has no subsequence which is convergent in uniform norm. (Hint: Show that every subsequence of {f} converges pointwise to some function. Can the subsequences converge uniformly?) (Remark: This is an example of a sequence of continuous functions bounded in the uniform norm which has no convergent subsequence.)