Theorem 9.22: Let {fn} be a sequence of functions defined on an interval I, and

Question

Theorem 9.22: Let {fn} be a sequence of functions defined on an interval I, and let x0 in I. If the sequence { fn } converges uniformly to some function f on I and if each of the functions fn is continuous at x0, then the function f is also continuous at x0. In particular, if each of the functions fn is continuous on I, then so too is f. Proof. Let in > 0. We must show there exists delta > 0 such that For each x in I we have Since uniformly. there exists N IN such that for all all n > N. We infer from inequalities (6) and (7) that We now use the continuity of the function fN. We choose delta > 0 such that if x in I and then, Combining (8) and (9), we have

Explanation / Answer

The prrof will not break down!!!

we can not take

1>=0>=x

It does not make any sense!!!

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