Double sequence, two sequences converge, but to different limits? [duplicate]
Solution 1:
A very simple example:
$$a_{m,n}=\begin{cases} 1,&\text{if }m\le n\\ 0,&\text{if }m>n\;. \end{cases}$$
If you write out the double sequence as an infinite array, it’s very easy to see what happens:
$$\begin{array}{ccc} 1&1&1&1&1&\ldots&\to&1\\ 0&1&1&1&1&\ldots&\to&1\\ 0&0&1&1&1&\ldots&\to&1\\ 0&0&0&1&1&\ldots&\to&1\\ 0&0&0&0&1&\ldots&\to&1\\ \vdots&\vdots&\vdots&\vdots&\vdots&\ddots\\ \downarrow&\downarrow&\downarrow&\downarrow&\downarrow&&&\vdots\\ 0&0&0&0&0&\ldots \end{array}$$
Solution 2:
$$a_{n,m}=\frac{n}{n+m}$$
One limit is 0 another is 1.