# Real analysis (Uniform continuous)

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Let {fn} be a sequence of continuous functions, all defined on [a,b]. Suppose {fn(a)} is a diverging sequence of real numbers. Prove that {fn} does not converge uniformly on (a,b]. (Notice a is not included in this interval.)

Assume {fn} converges unifromly and then use the diverging sequence of real numbers as a contradiction to f being continuous in the end.

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Proof:

Since is uniformly convergent on the interval , assume the limit function is , then for any , we can find some , ...

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The expert examines real analysis for uniform continuous.

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