Answer by Michael Hardy for Prove that if every PROPER sub sequence of a sequence is convergent , then the sequence is convergent

If every proper subsequence of $a_1,a_2,a_3,a_4,\ldots$ is convergent then $a_2,a_3,a_4,\ldots$ is convergent. But if $a_2,a_3,a_4,\ldots$ is convergent, then so is $a_1,a_2,a_3,a_4,\ldots$, and it has the same limit.