On existence of a convergent subsequence

Question

Let $(a_{(m,n)})_{m,n \in \mathbb{N}}$ be a double sequence of positive numbers.

Suppose that we know that there exists a limit $\lim_{m \to \infty}\lim_{n \to \infty}a_{(m,n)}=L$. Does there always exists an increasing function $m: \mathbb{N} \to \mathbb{N}$ such that $\lim_{n \to \infty}a_{(m(n),n)}=L$?

user66081 · Accepted Answer · 2015-01-30T18:59:26.383

1

I suppose you mean strictly increasing. Then the answer is no:

Take $a_{(m, n)} = 1$ if $m < n$, and $\ldots = 2$ if $m \geq n$. Then $L = 1$. Suppose $m : \mathbb{N} \to \mathbb{N}$ is strictly increasing; then $m(n) \geq n$, so that $\lim_{n \to \infty} a_{(m(n), n)} = 2$.

edited Jan 30 '15 at 18:59

answered Jan 29 '15 at 16:37

user66081

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In your example $(a_{(m,n)})_{m,n \in N}$ are not positive numbers. – Gogi Pantsulaia Jan 30 '15 at 14:04
You are correct. You can take under "zero" an arbitrary positive number which differs from 1. – Gogi Pantsulaia Jan 30 '15 at 15:31
Alright; corrected. – user66081 Jan 30 '15 at 18:58

On existence of a convergent subsequence

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