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Math Help - Limit proof

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    Limit proof

    Given that (x sub n) is a sequence of real numbers, that x > 0, and that (x sub n) approaches x as n approaches infinity, prove that there exists an integer N such that the inequality x > 0 holds for all integers n greater than or equal to N.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Slazenger3 View Post
    Given that (x sub n) is a sequence of real numbers, that x > 0, and that (x sub n) approaches x as n approaches infinity, prove that there exists an integer N such that the inequality x > 0 holds for all integers n greater than or equal to N.
    Let \frac{x}{2}=\varepsilon>0. By x_n\to x there exists some N\in\mathbb{N} such that N\leqslant n\implies |x_n-x|<\varepsilon\implies x-\varepsilon=\frac{x}{2}<x_n
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