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I am beginning proofs in analysis. I am reading Kane's book, but I am not sure if this proof counts as a proof in analysis. I have tried proving by contradiction, but I failed. I also tried using the triangle inequality ($\lvert x - 6\rvert\leq\lvert x\rvert + \vert\text{-}6\rvert$), but it just does not follow from it. The farthest I got was showing that $\lvert 2x -6\rvert\leq \rvert x\lvert + \rvert x - 6\lvert$ by letting $y = x-6$ and using the triangle inequality. If anybody has any idea how to proceed, I would greatly appreciate some guidance. Thanks.

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    $\begingroup$ Use triangular inequality: $a,b\in\mathbb{R}\;\implies\;|a+b|\le |a|+|b|$, and notice that $6=x+(-x+6)$ $\endgroup$ Oct 9 '18 at 22:43
  • $\begingroup$ @ÁngelMarioGallegos Thanks for your comment. It was what I needed to figure it out. $\endgroup$ Oct 9 '18 at 22:52
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Use the triangle inequality $$|a|+|b|\geq |a+b|$$ and the fact that $|-a|=|a|$.

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  • $\begingroup$ Thanks for your help. This part helped at the end of the proof. $\endgroup$ Oct 9 '18 at 22:57

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