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Math Help - Proof: Limits of Sequences

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    Proof: Limits of Sequences

    Supply a proof for the following theorem: Suppose that f is continuous at a and that x(subscript)n tends to a as n goes to infinity. Then there is an integer N such that f(x(subscript)n) is defined for all integers n > N; furthermore, f(x(subscript)n) tends to f(a) as n goes to infinity.

    Should I be typing these in LaTeX and posting them here somehow? I hope it is obvious that typing this out with regular script that x(subscript)n represents the sequence x-sub-n. This is my first post so I hope to clean up my future posts before submission to aviod confusion or ambiguity.
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    [tex]\lim_{n \to \infty}(x_n)\to a[/tex] gives \lim_{n \to \infty}(x_n)\to a
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by paulread View Post
    Supply a proof for the following theorem: Suppose that f is continuous at a and that x(subscript)n tends to a as n goes to infinity. Then there is an integer N such that f(x(subscript)n) is defined for all integers n > N; furthermore, f(x(subscript)n) tends to f(a) as n goes to infinity.

    Should I be typing these in LaTeX and posting them here somehow? I hope it is obvious that typing this out with regular script that x(subscript)n represents the sequence x-sub-n. This is my first post so I hope to clean up my future posts before submission to aviod confusion or ambiguity.
    Let N be any neighborhood of f(x), by f's continuity we know that f^{-1}(N) is a neighborhood of x and so by assumption all but finitely many of \left\{x_n\right\}_{n\in\mathbb{N}} and so all but finitely many of f(x_n) are in ff^{-1}(N)\subseteq N.

    The conclusion follows.

    P.S. You should probably specify what you kind of space we are in here. The above works for any topological space with a convergent sequence. Some weird things happen though. For example, any sequence in an indiscrete space converges to every point.
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    By the way, some textbooks define "f is continuous at x=a" to mean that \lim_{n\to\infty} f(x_n}= f(a) for every sequence \{x_n\} that converges to a.
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