Let $a_{m,n}= \frac{m}{m+n}.$ Compute the iterated limits $$\lim_{n\rightarrow \infty}(\lim_{m \rightarrow \infty} a_{mn})$$ and $$\lim_{m\rightarrow \infty}(\lim_{n \rightarrow \infty} a_{mn})$$

My attempt at solving the first limit is that if we take $(\lim_{m \rightarrow \infty} \frac{m}{m+n})$, for a fixed value of $n$ it should converge to $1$. And then $\lim_{n\rightarrow \infty} (1)=1.$ And similary for the the second limit if we take $(\lim_{n \rightarrow \infty} \frac{m}{m+n})$ for a fixed value of m, it converges to $0$ and then $\lim_{m\rightarrow \infty}(0)=0$.

Query 1: I haven't a good understanding of iterated limits as this is the first time I'm encountering the topic. Is my reasoning correct?

Query 2: Also Stephen Abbott doesn't cover iterated limits in his book but gives it as a difficult last question to a chapter (I'm guessing I'm supposed to struggle with this a while) so are there any other sources/book out there that cover iterated limits for sequences (not integrals)?A simple google search does not help.

  • $\begingroup$ You are correct. $\endgroup$ – RRL Jul 23 '18 at 23:51

There is a general theory that considers iterated limits in relation to double limits and, among other things, explains under what conditions

$$\tag{*}A=\lim_{n,m \to \infty} a_{mn} = \lim_{m\rightarrow \infty}(\lim_{n \rightarrow \infty} a_{mn}) = \lim_{n.\rightarrow \infty}(\lim_{m \rightarrow \infty} a_{mn}) $$

The general double limit is $A$ if for every $\epsilon > 0$ there exists a positive integer $N$ such that if $n,m > N$, then $|a_{mn} - A| < \epsilon$.

For example, if the double limit $\lim_{n,m \to \infty}a_{mn}$ exists and each of the single limits $\lim_{n \to \infty} a_{mn}$ and $\lim_{m \to \infty} a_{mn}$ exist for all $m$ in the first case and all $n$ in the second case, then (*) holds.

A good reference is The Elements of Real Analysis (2nd edition) by Bartle.


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