So Bolzano–Weierstrass gives you a convergent subsequence, with limitc, say. You also know that the whole sequence doesnotconverge toc.

Now try to write down what it means, in terms of the definition, to say that the sequence does not converge toc. The definition of a sequence converging tocsays that given any , the terms of the sequence eventually get within ofc. If that is not true, then there must exist some for which infinitely many terms of the sequence are at least distance away fromc. These terms have a convergent subsequence (by Bolzano–Weierstrass again), and because each term of that subsequence is at least distance away fromc, so is the limit of that subsequence.