Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$. Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$.
Here's my example: let $C$ = (0,1], where 1 is a limit point of $C$, let ($x_t$) = ($1/t$)=($1,1/2,1/3,...$), but ($x_t$) converges to zero instead of 1. 
Is my example correct? If not, could someone give another example please? Thanks.  

$1$ is not a limit point of $C$.
The definition for limit point is,($x$ of $C$)
$$ \forall r > 0, \exists y \in C\mid y \neq x, \operatorname{d}(x,y) < r
$$
so taking $r = 1/3$...
You can choose $x_j = 1/j$ for all even $j$ and $x_j = 1-1/j$ for all odd $j$. Then $0$ and $1$ are limit points of $C$, but the sequence does not converge.
