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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 $

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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.

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