Suppose that {a_n} is a divergent sequence. Show that it has a subsequence {a_{n_k}} with 1+ n_k < n_{k+1} which also diverges
can anyone give me a concise proof for this
You could try an argument by contradiction. Suppose the result is false, so that every such subsequence converges. In particular, the subsequences $\displaystyle a_{2n}$ and $\displaystyle a_{2n+1}$ (of even-numbered and odd-numbered terms terms in the sequence) both converge, say $\displaystyle a_{2n}\to l_1$ and $\displaystyle a_{2n+1}\to l_2$. The subsequence $\displaystyle a_{3n}$ also converges, and it's easy to see that its limit has to be both l_1 and l_2. So that means that $\displaystyle l_1=l_2$, and the whole sequence has to converge to that limit.