Converse of "positive sequence converges to positive limit"

It is well known, and not hard to prove, that if a convergent real sequence $(x_n)$ is eventually positive (i.e. there is some $K$ for which $n > K \Rightarrow x_n > 0$), then the limit to which it converges must be non-negative. I am interested in whether the converse is true: that is, if we know that $(x_n)$ converges to a non-negative limit, then is it true that there exists $K$ with the property stated above?

Let $(x_n)$ converges to $x(> 0)$. i.e.,

For a given $\epsilon>0, \hspace{1mm}\exists K\in \Bbb{N}$ , such that $\forall n\geq K, x_{n}\in (x-\epsilon, x+\epsilon)$. This works for any $\epsilon>0$.

Let the given $\epsilon:=x>0 \implies \exists K\in \Bbb{N}$ , such that $\forall n\geq K, x_{n}\in (x-\epsilon, x+\epsilon)=(0,2x)$ OR $0<x_{n}<2x$.

Hence, there exists a $K$ for which $x_{k}>0$ for all $n\geq K$.