Squeeze Theorem/Functions/Proof 2

Theorem
Let $a$ be a point on an open real interval $I$.

Also let $f$, $g$ and $h$ be real functions defined at all points of $I$ except for possibly at point $a$.

Suppose that:
 * $\forall x \ne a \in {I}: g \left({x}\right) \le f \left({x}\right) \le h \left({x}\right)$
 * $\displaystyle \lim_{x \to a} \ g \left({x}\right) = \lim_{x \to a} \ h \left({x}\right) = L$.

Then $\displaystyle \lim_{x \to a} \ f \left({x}\right) = L$.

Proof
Let $f, g, h$ be real functions defined on an open interval $\left({a \,.\,.\, b}\right)$, except possibly at the point $c \in \left({a \,.\,.\, b}\right)$.

Let:
 * $\displaystyle \lim_{x \to c} \ g \left({x}\right) = L$
 * $\displaystyle \lim_{x \to c} \ h \left({x}\right) = L$
 * $g \left({x}\right) \le f \left({x}\right) \le h \left({x}\right)$ except perhaps at $x = c$.

Let $\left \langle {x_n} \right \rangle$ be a sequence of points of $\left({a \,.\,.\, b}\right)$ such that $\forall n \in \N^*: x_n \ne c$ and $\displaystyle \lim_{n \to \infty} \ x_n = c$.

By Limit of Function by Convergent Sequences:
 * $\displaystyle \lim_{n \to \infty} \ g \left({x_n}\right) = L$

and:
 * $\displaystyle \lim_{n \to \infty} \ h \left({x_n}\right) = L$

Since:
 * $g \left({x_n}\right) \le f \left({x_n}\right) \le h \left({x_n}\right)$

it follows from the Squeeze Theorem for Sequences of Real Numbers that:
 * $\displaystyle \lim_{n \to \infty} \ f \left({x_n}\right) = L$

The result follows from Limit of Function by Convergent Sequences.