Chinese remainder theorem and Wikipedia Proof

Question

I'm looking at the proof

https://en.wikipedia.org/wiki/Chinese_remainder_theorem#Existence_(direct_construction)

for the existence in the Chinese remainder theorem.

Isn't the proof that $\;0\le x\le N$ missing? How do I prove it starting from $$ x=\sum _{i=1}^{k}a_{i}M_{i}N_{i}? $$

See the link for the notation.

Answer 1

That number is a solution of the system of congruences, but it doesn't have to be smaller than $N$ indeed. After having computed it, you simply replace it by its remainder after dividing it by $N$.