I would like to provide a little more detail and offer the proof for the infinitude of the primes. Suppose that there was a last prime number. If we numbered the primes, we'd have $p_1$, $p_2$ and so on until the last one which we'll call $p_k$.
We now consider the product of all of these prime numbers plus 1. $n=p_1p_2p_3\cdots p_k+1$. To check to see if a number is prime, it suffices to divide by all prime numbers smaller than it, so if we divide out any of the primes, maybe $p_i$ for all $i$ between $1$ and $k$, we would have then ${p_1p_2\cdots p_{i-1}p_ip_{i+1}\cdots p_k$+1} \over p_{i}$. We could simplify this term then to include a product of primes on the left, cancelling out $p_i$, but then we would still have the last term $1\over{p_i}$, which is not an integer since $p_i$ is greater than 1. But that means that $n$ was not divisible by $p_i$. Since $i$ was any arbitrary positive integer less than or equal to $k$, $n$ is not divisible by any of the primes smaller than itself, so $n$ too is prime, contradicting the idea of a last prime.
So we conclude their are infinitely many primes.