Squeeze Theorem/Functions/Proof 2

Proof
Let $f, g, h$ be real functions defined on an open interval $\openint a b$, except possibly at the point $c \in \openint a b$.

Let:
 * $\ds \lim_{x \mathop \to c} \map g x = L$
 * $\ds \lim_{x \mathop \to c} \map h x = L$
 * $\map g x \le \map f x \le \map h x$ except perhaps at $x = c$.

Let $\sequence {x_n}$ be a sequence of points of $\openint a b$ such that:
 * $\forall n \in \N_{>0}: x_n \ne c$

and:
 * $\ds \lim_{n \mathop \to \infty} \ x_n = c$

By Limit of Function by Convergent Sequences:
 * $\ds \lim_{n \mathop \to \infty} \map g {x_n} = L$

and:
 * $\ds \lim_{n \mathop \to \infty} \map h {x_n} = L$

Since:
 * $\map g {x_n} \le \map f {x_n} \le \map h {x_n}$

it follows from the Squeeze Theorem for Real Sequences that:
 * $\ds \lim_{n \mathop \to \infty} \map f {x_n} = L$

The result follows from Limit of Function by Convergent Sequences.