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

Question

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,

Answer 1

Note that by definition

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