Let $(x_n)_{n \geq 1}$ be a sequence of real numbers. Suppose that we are able to show that for a fixed number $m$, $(y_n)_{n \geq 1}:= (x_{n+m})$ and we know that $\lim_{n\to\infty}(y_n)=x$ for some $x \in \mathbb R$. Then is it true that $\lim \sup_{n \rightarrow \infty}(x_n)$ exists and is equal to $x$?
I think that this is true because $\lim_{n\to\infty}(y_n)=x$ implies $\lim_{n\to\infty}(x_n)=x$ (since in case of limits one is allowed to ignore first finitely many terms) an therefore, $\lim \sup_{n \rightarrow \infty}(x_n)= \lim_{n\to\infty}(x_n)=x$.
Please correct me if this is wrong. Any remark is appreciated.