# Sequence with every proper subsequence converges but the sequence doesn't [closed]

Here by proper subsequence I mean a subsequence which leaves out infinitely many indices. For example $$1/2,1/3, \ldots$$ is not a proper subsequence of $$1,1/2, \ldots$$

This is impossible. Let's say $$a_n$$ doesn't converge, but every proper subsequence converges. Then the sequences $$a_{2n}$$, $$a_{2n+1}$$ converge, so in order for $$a_n$$ not to converge we need $$a_{2n}\to L_1$$, $$a_{2n+1}\to L_2$$ with $$L_1\neq L_2$$. Now consider the sequence $$a_{3n}$$. The subsequence $$a_{6n}$$ converges to $$L_1$$ and $$a_{6n+3}$$ to $$L_2$$, so $$a_{3n}$$ does not converge, which is a contradiction.