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What are the sufficient conditions of interchanging two limits of a double sequence? I found some answers here: When can you switch the order of limits?

However, I have the following example which seems to satisfy a set of sufficient conditions (e.g., for sufficiently large $n_{0}$ $a_{n_{0},m}$ converges and $\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} a_{n,m}=L$), but the limits are different.

The example is: Let $X_n$ be a sequence of ${\tt Uniform}(0.5−n^{−1},0.5+n^{−1})$. Define $a_{n,m}=F_{n}(0.5+m^{−1})$, where $F_n$ is the distribution function of $X_n$. Clearly, $\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} a_{n,m}=0.5$ but $\lim_{m\rightarrow\infty}\lim_{n\rightarrow\infty} a_{n,m}=1$.

I am not sure where I am wrong. It will be helpful if I can clear my understanding.

user2660120
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  • I do not understand the notation you are using. But the post you links to has another important assumption, which is that the double sequence satisfies that for all $\epsilon\gt 0$ there exist and $N$ such that if $n,m\geq N$, then $|a_{nm}-L|\lt \epsilon$. You never mention that condition. Did you check it? – Arturo Magidin Nov 09 '21 at 06:28
  • Thanks! So, that means we have to find the limit w.r.t. the double sequence first. Does it mean that we need to first check if limits w.r.t. both the orderings are the same? – user2660120 Nov 09 '21 at 07:39
  • No, it does not mean that. If you actually read the post you link to carefully, you will see that the limit as described in the above comment is a very strong type of convergence, and that it is independent of the two iterated limits being the same. The posts there give some sufficienty conditions for the two iterated limits being the same, but they are not necessary and you can't do things in whatever order you want to get to a conclusion. – Arturo Magidin Nov 09 '21 at 13:44

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