By applying the sequential criterion for functional limits to prove the squeeze theorem:
Let $f,g,h$ be functions on a common domain $A \subset \mathbb{R}$ such that for each $x \in A$ , $f(x) \leq g(x) \leq h(x)$. If $c$ is a limit point of $A$ with $$\lim_{x \rightarrow c}{f(x)}=\lim_{x \rightarrow c}{g(x)}=L$$, then $\lim_{x \rightarrow c}{g(x)}$ exists and $$\lim_{x \rightarrow c}{g(x)}=L$$ Can anyone guide me ? I don know how to use sequential criterion to prove this.
[Math] By applying the sequential criterion for functional limits to prove the squeeze theorem
real-analysis
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Best Answer
You know that $$f(x)\leq g(x) \leq h(x)$$ So $$\lim_{x\to c} f(x) \leq \liminf_{x\to c} g(x)\leq \limsup_{x\to c} g(x) \leq \lim_{x\to c} h(x)$$ so $$L \leq \lim_{x\to c} g(x) \leq L$$ You use at first the limsup and liminf as the lim doesn't need to exist (you are showing it exists) but limsup and liminf always exists (when you allow $\pm \infty$)