  This paper contains two improvements on a theorem of S. N. Bernstein for
Banach spaces. We show that if $X$ is an arbitrary infinite-dimensional Banach
space, $\{Y_n\}$ is a sequence of strictly nested subspaces of $ X$ and if
$\{d_n\}$ is a non-increasing sequence of non-negative numbers tending to 0,
then for any $c\in(0,1]$ we can find $x_{c} \in X$, such that the distance
$\rho(x_{c}, Y_n)$ from $x_{c}$ to $Y_n$ satisfies $$ c d_n \leq
\rho(x_{c},Y_n) \leq 4c d_n,~\mbox{for all $n\in\mathbb N$}. $$ We prove the
above inequality by first improving Borodin (2006)'s result for Banach spaces
by weakening his condition on the sequence $\{d_n\}$. The weakened condition on
$d_n$ requires refinement of Borodin's construction to extract an element in
$X$, whose distances from the nested subspaces are precisely the given values
$d_n$.
