Hi guys could you please help me on this question I'm confused.
question: when does the equality hold in the triangle inequality:
my attempt :
$|x + y| \leq |x| + |y|$ this implies
$(|x+y|)^2 = (|x| + |y|)^2 \Rightarrow (x+y)^2 = (|x| + |y|)^2 \Rightarrow x^2 + 2xy + y^2 = |x|^2 + 2|x||y| + |y| ^2 $
since $x^2 = |x|^2$
$\Rightarrow 2xy = 2|x||y| \Rightarrow xy = |x||y| $
and $|x \cdot y| = |x| \cdot |y| \text{ iff } x = y$
therefore $xy = |xy| \Rightarrow xx = |xx| \Rightarrow x^2 = |x|^2 \Rightarrow x=x $
that is what I did so far, I wanted to know if I did this write or if there is a better way in showing this inequality and also what conditions does it hold? I am really confused on that one. This question was on our past quizzes we have a quiz coming up so I wanted to do all the questions and be prepared.
Thank you in advance