[Math] Is the set of all prime numbers bounded

Question

I'm a bit confused about this.

I know that there is an infinite amount of prime numbers, a set having infinitely many elements does not necessarily make that set bounded. For example, the set $A$ of all real numbers between $1$ and $2$ has an infinite amount of elements, but is bounded clearly (because there exist numbers $C$ and $D$ such that $C<a<D$ for all $a\in A$.

Here is my attempt to explain that the set of all prime numbers is not bounded.

Let $N$ be the set of all prime numbers. We know that there are an infinite amount of prime numbers. Thus we know that there does not exist a number $R$ such that $n<R$ for all $n \in N$.

But do we know that?

According to Euclid's theorem of infinite primes, the proof just states that there is always at least 1 prime number in the set of all primes, and therefore there is an infinite amount of prime numbers. However, it explicitly states that what is found in this proof is another prime–one not in the given initial set. There is no size restriction on this new prime, it may even be smaller than some of those in the initial set.

So is the set of all prime numbers bounded? If so, how do I explain that.

Any help would be appreciated.

Answer 1

An infinite collection of natural numbers is always unbounded.