## If $\left\{x_n\right\}$ is a convergent sequence of points in $[a, b]$ and $lim x_n = c$, then $c\in[a, b]$

If $$\left\{x_n\right\}$$ is a convergent sequence of points in $$[a, b]$$ and $$lim x_n = c$$, then $$c\in[a, b]$$.

This is a statement that I found in my real analysis text book. How can I prove the above? Should I use the theorem:

If $$\left\{x_n\right\}$$ and $$\left\{y_n\right\}$$ are two convergent sequences and there exists a natural number $$m$$ such that $$x_n>y_n$$ for all $$n\geq m$$, then $$lim x_n\geq lim y_n$$.