Problem:

Suppose that is a sequence of points in that converges to the point u and that . Prove that there is an indexKsuch that if

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Attempt:

By given, and .

I believe I have to use following criterion, but do not know how to apply it to prove the problem.

Definition:A sequence of points in is said to converge componentwise to the point u for each indexiwith , then

.

Componentwise Convergence Criterion:Let be a sequence in and let . Then converges to u if and only if converges componentwise to u.

Thank you for your help.