Let {${a_n}$} be sequence of real number.
I understand the the fact that
(ⅰ) {${a_n}$} converges to $a$ when $n→∞$
(ⅱ)For arbitrary strictly increasing sequence of natural numbers ${n(i)}$, there exists strictly increasing sequence of natural numbers ${n(j)}$, such that {${a_n(i(j))}$} converges to $a$ when $j→∞$.
My question : What kind of benefit from characterizing congvergence like (ⅱ)? Also, I have never seen this characterization in any reference. If you could find any refference about this, I would be grateful if you could tell me.
