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I am currently learning topology from Munkres. The question below is an exercise in section 45.

Let $f_n\colon I\to \mathbb{R}$ be the function $f_n(x)=x^n$. The collection $F=\{f_n\}$ is pointwise bounded but the sequence $(f_n)$ has no uniformly convergent subsequence; at what point or points does $F$ fail to be equicontinuous.

Any help would be appreciated. Thank You.

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    Can you find any point in $I$ where the family of functions _is_ equicontinuous?2012-04-13
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    It might help to think about what the limit function is.2012-04-13
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    hint: consider the point on the right hand side of the interval $I$ (1). Can you find the open set required by equicontinuity for an $\epsilon<1$ for ALL functions in $F$?2012-04-13

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