Is $a=b$ when $|a-b|=0$? If so, how can I prove it?

Question

I was looking at another proof in my book which seemed to use this fact for granted. I am pretty sure it's right, but how could I prove that?

Answer 1

The definition of absolute value is

$$ |y|=\begin{cases} y &\text{if }y\geq0\\ -y &\text{if }y\leq0\end{cases} $$

Therefore, if $|y|=c$, it follows that either $y=c$ or $-y=c$. Can you finish the proof from here?

Answer 2

If you wish, you can write two equations: if $|a-b|=0$, then either $$ a - b = 0, $$ or $$ b - a = 0, $$ which means $$ a = b, $$ or $$ b = a $$

Answer 3

Well, your question seems to have a very fundamental answer.

$1. $ If you are aware of Normed Linear Spaces then the answer follows from the definition of norm $|\cdot|$ of $\Bbb R$.

$2.$ If you do not like the answer $1.$ then recall that $|x|=\begin{cases} x,\text{ if $x\ge0$}\\ -x,\text{ if $x< 0$}\end {cases}$

Here given that $|a-b|=0\implies a-b=0\text{ or }b-a=0$ in both case $a=b$.