How can I prove the following? $|x + x_1 + x_2 + ... + x_n| ≥ |x| − (|x_1| + |x_2| + ...+|x_n|)$

Question

Let $x$ be real, and $n$ is natural. How do I get started of proving such thing?

Answer 1

I'm assuming you are considering real numbers (this inequality holds for more general cases). If you know the triangle inequality for real numbers $|a+b|\le |a|+|b|$ then $$|a| = |a+b-b|\le |a+b|+|b| \Rightarrow |a|-|b| \le |a+b|.$$ Now your result follows by induction.