Let $X$ be separable Banach space and $\{x_n\}$ be a bounded sequence, relatively weakly compact sequence in $X$. we set $y_n=\frac{1}{n}\sum_{i=1}^{n}{x_i}$, then (together with the Krein and Eberlein-Smulian theorems), we can assume that there exists a **subsequence** of $\{y_n\}$ converges weakly to some element $y \in X$.

We suppose that  every **weak limit** point of $\{y_n\}$ must equal $y$. 

Can we say that $\{y_n\}$ must converge weakly to $y$?