Consider a sequence of real numbers $\{X_n\}_{n \in \mathbb{N}}$ and suppose that I have shown that
$\forall \epsilon>0$, $\limsup_{n\rightarrow \infty} X_n-X\leq\epsilon$ and $\liminf_{n\rightarrow \infty} X_n-X\geq-\epsilon$, $X \in \mathbb{R}$
Questions:
1) Does this imply $\lim_{n \rightarrow \infty} X_n=X$?
2) Why yes or not?
The relations $\limsup X_n-X\leq\varepsilon$ and $\liminf X_n-X\geq-\varepsilon$ imply $$\limsup X_n-\liminf X_n\leq2\varepsilon$$ for all $\varepsilon>0$; thus $X=\limsup X_n=\liminf X_n$.