In lecture in my real analysis course the other day we were proving that absolute convergence of a series implies convergence. Our professor started off by showing us the(adsbygoogle = window.adsbygoogle || []).push({}); wrongway to prove it:

[tex]\left| \sum_{k=1}^\infty a_k \right| \leq \sum_{k=1}^\infty \left| a_k \right| < \epsilon[/tex]

Then he demonstrated the correct proof, by showing that the sequence of partial sums is Cauchy convergent and then using the triangle inequality.

But this got me thinking: why is the first proof wrong? I definitely agree that the second proof is more solid, but if the triangle inequality is proved by induction, meaning it's true for all natural numbers, isn't that, well, infinite? I was wondering if someone could supply a counterargument or proof by contradiction illustrating why this conclusion is incorrect. Thanks as always.

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# Triangle inequality with countably infinite terms

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