Please help me to solve a problem given in the survey Minimal Idempotents and Ergodic Ramsey theory by Vitaly Bergelson(Exercise 15(iii), page 23), which is
Problem: Prove that if $x_1,x_2$ are proximal points in a topological system $(X,T)$($X$ is compact and $T$ is continuous), then there exists $p\in \beta\mathbb{N}$ such that $p-lim_{n\in \mathbb{N}}T^nx_1=p-lim_{n\in \mathbb{N}}T^nx_2.$
Definition(Proximal point): Given a dynamical system $(X,T),x_1,x_2\in X$ is called proximal if there exists a sequence $n_k\rightarrow \infty$ such that $d(T^{n_k}x_1,T^{n_k}x_2)\rightarrow 0.$
Definition(p-limit): Given an ultrafilter $p\in \beta \mathbb{N}$ and a sequence $\{x_n\}_{n\in \mathbb{N}}$ in a topological space $X$, one writes $p-lim_{n\in\mathbb{N}}=y$ if for every neighbourhood $U$ of $y$, one has $\{n:x_n\in U\}\in p.$