Given two closed convex sets $A,B\subset\mathbb{R}^n$, with $A\cap B=\varnothing$. Consider the alternating projection algorithm with arbitrary initial point $x_0$ and update rule: $y_k=P_A(x_{k-1})$ and $x_k=P_B(y_k)$ for each iteration $k$. Here $P_C(x)$ denotes the projection of $x$ onto a closed convex set $C$, i.e., $P_C(x)=\arg\min_{y\in C}\;\frac{1}{2}\lVert x-y\rVert_2^2$. How to show $\lVert x_k-y_k\rVert_2\to dist(A,B)\triangleq\inf_{x\in A,y\in B}\lVert x-y\rVert_2$ by using some elementary arguments in optimization or mathematical analysis?
I know that in some paper, a more general version of this statement for Hilbert space is given, but the proof is quite complicated. Here the space is of finite dimension, so do we have some easier method to prove it, e.g., this is a question for some PhD level course?