The problem is: \"Show that if {a(n)} converges to L and if {a(n)} <= L for infi
ID: 2961425 • Letter: T
Question
The problem is: "Show that if {a(n)} converges to L and if {a(n)} <= L for infinitely many values of n, then there is a subsequence of {a(n)} that is increasing and converges to L."
My teacher recommended that we consider two cases:
Case I: a(n) = L for infinitely many n.
Case II: a(n) = L for at most finitely many n and so a(n) < L for infinitely many n.
I am very confused by this problem, since we didn't cover subsequences in class. I'd appreciate it if anyone could explain this problem to me. I will quickly give the points to the best answer. Thanks!
Explanation / Answer
Suppose a(i) is the minimum element of {a(n)}.
take a(i) as the first element of subsequence.
If we remove the first i elements from sequence,we still have a sequence with infinite elements less than equal to L. take the minimum(say a(j)) among those elemnts as second term of subsequence. We repeat this process after removing first j terms of sequence. continuing this process we get an increasing subsequence converging to L
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