Squeeze Theorem

Theorem: Suppose that $$X=(x_n), Y=(y_n),\,$$ and $$Z=(z_n)\,$$ and that $$x_n \le y_n \le z_n$$ $$\forall n \in \textbf{N}.$$ Then $$Y\,$$ is convergent and $$\lim_{n\rightarrow\infty}{X} = \lim_{n\rightarrow\infty}{Y} = \lim_{n\rightarrow\infty}{Z}$$.

Direct Proof
Let $$w = \lim_{n\rightarrow\infty}{X} = \lim_{n\rightarrow\infty}{Z}$$. If $$\varepsilon > 0\,$$ is given, then it follows from the convergence of $$X\,$$ and $$Z\,$$ to $$w\,$$ that there exists a natural number $$K\,$$ such that if $$n \ge K$$ then $$|x_n - w| < \varepsilon\,$$ and $$|z_n - w| < \varepsilon\,$$. Since $$x_n \le y_n \le z_n$$ by hypothesis, we have $$x_n - w \le y_n - w \le z_n - w$$, with $$z_n - w < \varepsilon\,$$ and $$-\varepsilon < x_n - w\,$$. It follows, then, that $$-\varepsilon < y_n - w < \varepsilon\,$$ $$\forall n \ge K$$. Since $$\varepsilon > 0\,$$ is arbitrary, it follows that $$\lim_{n\rightarrow\infty}{Y} = w$$ as desired.

QED