Consider the following proposition:
Proposition 1. Let $X$ be a reflexive Banach space and suppose that $\{x_n\}$ is a sequence in $X$ that is bounded and has at most one weakly sequentially cluster point $x$. Then $\{x_n\}$ converges weakly to $x$.
We have known that Proposition 1 is a direct corollary of the following theorems and proposition:
Theorem 1. (Kakutani) Let $X$ be a Banach space. Then $X$ is reflexive if and only if $$ B_X=\{x\in X: \|x\|\leq 1\} $$ is compact in the weak topology $\sigma(X, X^*)$.
Theorem 2. (Eberline-Smulian) Let $X$ be a Banach space. Then the following statements are equivalent:
(i) $B_X$ is weakly compact;
(ii) $B_X$ is sequentially weakly compact.
Proposition 2. Let $C$ be a sequentially compact subset of a Hausdorff space $X$ and suppose that $\{x_n\}$ is a sequence in $C$ that admits a unique sequential cluster point $x$. Then $\{x_n\}$ converges to $x$.
I woulk like to ask where we can find the reference of Proposition 1.
Thank you for all comments and helping.