Proof of Infinite Primes

In one of my earlier posts, I mentioned how
since there are an infinite number of primes,
no number system can exist that eliminates
all repeating decimals. I didn't elaborate on how
I knew that, so I decide to put the proof here.
By the way, this isn't my proof.

Suppose there are a limited number of primes,
and the largest of them is say, n.
Now, multiply all the primes, including n.
This operation will result in a number p.
Add 1 to p. This new number cannot be divided
by any of the primes in the old list, meaning that
if it is a prime, then we have a new number added
to our list. If it isn't, then there will have to be
another prime which wasn't in our list that divides
into our new number. Either way, we have a
new prime added to our list. Continue this process,
and we have an infinite number of primes.