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$S(n,m)$ is a double sequence. Can anyone give me an example where lim$_{m , n \to \infty} S(n,m)$ exists but lim$_{n \to \infty}$( lim$_{m \to \infty} S(n,m)$) , lim$_{m \to \infty}$( lim$_{n \to \infty} S(n,m)$) do not?

My Attempt: I thought of an example.

$S(1 ,m) =m $, $S(n , 1) =n $,

$S(n,m) = 1 $ otherwise.

But this does not seem to be a good example . As we ignore the first row and first column , two iterated limits exist. I want an example where lim$_{m \to \infty} S(n,m)$ , lim$_{n \to \infty} S(n,m)$ will not exist for infinitely many $n$ and $m$ respectively.

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1 Answer 1

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You had the example $\frac {(-1)^ m}{n}$, showing that one of the iterated limits did not exist. Using that we can make an example where both iterated limits fail: $$ \frac {(-1)^ m}{n} + \frac {(-1)^ n}{m} $$ The idea is that $\lim_{m, n\to\infty}$ looks at the whole region of the plane containing the points $(m, n)$ with $m, n>N$ for increasing $N$, and seing whether they all eventually look similar under $S$. In this case, all elements have absolute value below $\frac2N$. The iterated limits, on the other hand, take limits along vertical lines in the plane (or horizontal, depending on the lmit order), and then looks at those limits as you take lines further and further from the origin. None of these lines have a limit, so the resulting limit does not exist.

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