Proof of series inequality using triangle inequality

Question

Prove the following inequality:

$$|x+x_1+x_2+...+x_n|\ge|x|-(|x_1|+|x_2|+...+|x_n|)$$

I've spent a long time on this one. I've come up with something but I don't think it's correct.

According to the triangle inequality $|x+y|\le|x|+|y|$:

$$|x_1+x_2+...+x_n|\le|x_1|+|x_2|+...+|x_n|$$

So then

$$|x|+|x_1|+|x_2|+...+|x_n|\ge|x|-(|x_1|+|x_2|+...+|x_n|)$$ $$\Downarrow$$ $$|x_1|+|x_2|+...+|x_n|\ge-|x_1|-|x_2|-...-|x_n|$$

Which is supposed to be proof of the first statement according to some people here but I think we only proved that a bigger number than the one in the L.H.S. is bigger than the one in the R.H.S., which doesn't mean that the L.H.S. is bigger than the R.H.S.

Thanks for your help.

Answer 1

Using the triangle inequality, $$|x|=|x+x_1+\cdots +x_n-(x_1+\cdots + x_n)|\le $$ $$\le|x+x_1+\cdots+x_n|+|-x_1|+\cdots +|-x_n|=$$ $$=|x+x_1+\cdots+x_n|+|x_1|+\cdots +|x_n|$$