My question is if 0 ≤ a, b < n and a congruent to b mod n, then how is a = b?

Question

I was trying to understand how this works in order to write a proof and I am unable to.

Answer 1

$b-a$ is a multiple of $n$, yet $b-a\le b-0<n$ and $b-a> b-n\ge 0-n$. Therefore $b-a=0$.

Answer 2

$a\equiv b \pmod n$ means $n|a-b$; i.e., $a-b=kn$.

If $0\le a<n$ and $0\le b < n$ (i.e., $-n<-b\le0$) then $-n\lt a-b <n$, so $k=0$, so $a-b=0$.

Answer 3

By definition $n$ divides $a-b$, hence $|b-a|$. However $\;0\le |b-a|<n$ if $0\le a,b<n$. How many nonnegative multiples of $n$ which are smaller than $n$ do you know?