Bernstein Lethargy Theorem and Reflexivity

In this paper, we prove the equivalence of reflexive Banach spaces and those Banach spaces which satisfy the following form of Bernstein's Lethargy Theorem. Let $X$ be an arbitrary infinite-dimensional Banach space, and let the real-valued sequence $\{d_n\}_{n\ge1}$ decrease to $0$. Suppose that $\{Y_n\}_{n\ge1}$ is a system of strictly nested subspaces of $X$ such that $\overline Y_n \subset Y_{n+1}$ for all $n\ge1$ and for each $n\ge1$, there exists $y_n\in Y_{n+1}\backslash Y_n$ such that the distance $\rho(y_n,Y_n)$ from $y_n$ to the subspace $Y_n$ satisfies $$\rho(y_n,Y_n)=\|y_n\|.$$ Then, there exists an element $x\in X$ such that $\rho(x,Y_n)=d_n$ for all $n\ge1$.