Unique subsequential limit of a bounded sequence
= Unique subsequential limit of a bounded sequence
If a bounded real sequence has the property that every convergent subsequence converges to the same number $L$, then the full sequence converges to $L$. Otherwise, a subsequence stays at least some fixed distance from $L$; Bolzano--Weierstrass gives it a convergent subsubsequence, contradicting the assumed uniqueness.