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$ .

Please anyone help me. Thanks in advance.