Let $X$ is a metric space with metric d and $f:X\to X$ is a continuous map.we call $(X,f)$ is dynamical system.dynamical system $(X,f)$ is topological transitive if for any pair of nonempty open sets $U,V\subseteq X$, there is an $n\in \mathbb{Z}_{+}$ such that $f^n(U)\cap V\neq \emptyset$. Recall that for $\delta > 0$ and $x,y \in X$, a $\delta$-chain of $f$ from $x$ to $y$ of length $n\in\mathbb{N}$ is a finite sequence $\{x_{0}=x,x_{1},...,x_n=y\}$ satisfying $d(f(x_i),x_{i+1})<\delta$ for $0\leq i\leq n-1$. $(X,f)$ is said to be chain transitive if for any $\delta>0$ and any pair $x,y\in X$, there is a $\delta-$chain of $f$ from $x$ to $y$.
Lemma:A topological transitive is chain transitive. please proof this lemma.