Problem:
Suppose that is a sequence of points in that converges to the point u and that . Prove that there is an index K such 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 index i with , 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.