# sequence

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• Oct 30th 2009, 12:56 PM
flower3
sequence
Prove that if $x_n \to x$ and $x>0$ , then there is $k \in N$ such that $\frac {x}{2} < x_n < \frac{3x}{2}, \forall n \geq k$
• Oct 30th 2009, 01:21 PM
Bruno J.
By definition, since $x_n$ converges to $x$, we can find $k$ such that $|x_n-x|<\frac{x}{2}$ whenever $n\geq k$, i.e.

$\frac{-x}{2}

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

$\frac{x}{2}

whenever $n\geq k$.