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This is a proof that my teacher gave I'm having a hard time with the last line of the proof.

Suppose $(a)_{j=1}^{\infty}$ is a sequence of real numbers. prove by induction on n that $$|\sum_{j=1}^{n}a_j|\leq\sum_{j=1}^{n}|a_j|$$ Induction step: $$|\sum_{j=1}^{n+1}a_j|=|a_{n+1}+\sum_{j=1}^{n}a_j|$$$$\leq|a_{n+1}|+|\sum_{j=1}^{n}a_j|$$$$\leq|a_{n+1}|+\sum_{j=1}^{n}|a_j|$$$$=\sum_{j=1}^{n}|a_j|$$

I don't know why $$|a_{n+1}|+\sum_{j=1}^{n}|a_j|=\sum_{j=1}^{n}|a_j|$$

Any help would be greatly appreciated.

Thanks,

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  • $\begingroup$ There is a typo. The right sum's upper index is $n+1$. $\endgroup$ – max_zorn Apr 14 '18 at 20:14
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    $\begingroup$ So the last line should be $|a_{n+1}|+\sum_{j=1}^{n}|a_j|=\sum_{j=1}^{n+1}|a_j|$?@max_zorn $\endgroup$ – jack Apr 14 '18 at 20:20
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Note that by definition

$$\sum_{j=1}^{n}|a_j|+|a_{n+1}|=\sum_{j=1}^{\color{red}{n+1}}|a_j|$$

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