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Math Help - Real Analysis - Functional Limits #2

  1. #1
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    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.
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  2. #2
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    Given \epsilon>0, since f\to\ell and h\to\ell as x\to c, it follows that \ell-\epsilon<f<\ell+\epsilon and \ell-\epsilon<h<\ell+\epsilon. Besides, as x closes to c, it's f\le g\le h, then \ell-\epsilon<f\le g\le h<\ell+\epsilon, hence |g-\ell|<\epsilon, therefore g\to\ell as x\to c.\quad\blacksquare
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