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Topological limit of a function

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Let X be a subset of R, let E be a subset of X, let x_0 be an adherent point of E, and let f: X-->R, g: X-->R, h: X-->R be functions such that f(x) <= g(x) <= h(x) for all x belonging to E.

If we have lim of x-->x_0; x belonging to E of f(x) = lim of x-->x_0; x belonging to E of h(x) = L for some real number L, show that lim of x-->x_0; x belonging to E of g(x) = L.

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Solution Summary

We prove a generalized theorem for functions converging to a limit when their domain converges to a specific point.

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