Let $\{u^k\}\subset \mathbb{R}^n$ be a sequence such that there exists a subsequence $\{u^{k_i}\}\subset \{u^k\}$ converging to $\bar{u}\in \mathbb{R}^n$.
I would like to ask when we have a stronger conclusion that $\{u^k\}$ converges to $\bar{u}$. For example, if $\{\|u^k-\bar{u}\|\}$ is monotonically decreasing then $\{u^k\}$ converges to $\bar{u}$.