Math Help - Real Analysis - Functional Limits #2

1. Real Analysis - Functional Limits #2

(Squeeze Theorem)
Let f,g, and h satisfy f(x) <= g(x) <= h(x) for all x in some common domain A. If lim as x-->c of f(x) = L and lim as x-->c of h(x) = L at some limit point c of A, show lim as x-->c g(x) = L as well.

2. Given $\epsilon>0,$ since $f\to\ell$ and $h\to\ell$ as $x\to c,$ it follows that $\ell-\epsilon and $\ell-\epsilon Besides, as $x$ closes to $c,$ it's $f\le g\le h,$ then $\ell-\epsilon hence $|g-\ell|<\epsilon,$ therefore $g\to\ell$ as $x\to c.\quad\blacksquare$