A (topological) transitive point must be a recurrent point?

Question

Given a continuous map $f:X\to X$ (maybe $X$ is a metric space, even $X$ is a compact metric space), the point $x$ is a transitive point, if $x$'s orbit is dense in $X$. And $x$ is a recurrent point if there exists a subsequence of $x$'s orbit such that this subsequence converges to $x$.

My question is that: A (topological) transitive point must be a recurrent point?

Answer 1

Consider $X = \{1/n: n \in \mathbb{N}\}\cup \{0\}$ and $f:X\to X$ given by $f(0)=0$, $f(1/n)=1/(n+1)$. $X$ is compact; $f$ is continuous; $1$ has a dense orbit but $1$ is not recurrent (not even nonwandering).