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Thread: sequence

  1. #1
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    sequence

    Prove that if $\displaystyle x_n \to x $ and $\displaystyle x>0 $ , then there is $\displaystyle k \in N $ such that $\displaystyle \frac {x}{2} < x_n < \frac{3x}{2}, \forall n \geq k $
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    MHF Contributor Bruno J.'s Avatar
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    By definition, since $\displaystyle x_n$ converges to $\displaystyle x$, we can find $\displaystyle k$ such that $\displaystyle |x_n-x|<\frac{x}{2}$ whenever $\displaystyle n\geq k$, i.e.

    $\displaystyle \frac{-x}{2}<x_n-x<\frac{x}{2}$

    whenever $\displaystyle n\geq k$. Adding $\displaystyle x$ throughout this inequality we get

    $\displaystyle \frac{x}{2}<x_n<\frac{3x}{2}$

    whenever $\displaystyle n\geq k$.
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