Deviation of elements of a Banach space from a system of subspaces

We prove that if $X$ is a real Banach space, $Y_1\subset Y_2\subset\dots$ is a sequence of strictly embedded closed linear subspaces of $X$, and $d_1\ge d_2\ge\dots$ is a nonincreasing sequence converging to zero, then there exists an element $x\in X$ such that the distance $\rho(x,Y_n)$ from $x$ to $Y_n$ satisfies the inequalities $d_n\le\rho(x,Y_n)\le8d_n$ for $n=1,2,\dots$.