Prove the triangle inequality I want to porve the triangle inequality:  $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$  I figured out that probably the cases:   $x\geq0$ and $y \geq 0$ $x&lt;0$ and $y &lt; 0$ $x\geq0$ and $y &lt; 0$ $x&lt;0$ and $y \geq 0$ &lt;- Here I am not sure...   have to be proven. However, I do not figured out a concrete method. Are my assumptions true?  How to finish the prove with these assumptions. I really appreciate your answer!!!
Proof of triangle inequality I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
how do I prove that $1 > 0$ in an ordered field? I've started studying calculus.  As part of studies I've encountered a question. How does one prove that $1 &gt; 0$?  I tried proving it by contradiction by saying that $1 &lt; 0$, but I can't seem to contradict this hypothesis.  Any help will be welcomed.