Help! Can you please see if there are any errors in this proof by contradiction that I wrote? Assume that there exists a positive integer n with a prime factorization that is not unique up to the ordering of its prime factors. This means there must at least two different prime factorizations for n.
Let $n = p_1 \cdot p_2 \cdots p_m = q_1 \cdot q_2 \cdots q_k$, where each $p_i$ and $q_j$ are positive prime numbers and each $p_i \neq q_j.$ Without loss of generality, assume that $p_1 \leq p_i$ and that $p_1 < q_j.$ Since $p_1$ is a factor of $n$, $p_1 \mid n.$ However, since $p_i \neq q_j$ implies $p_1 \neq q_j,$ it must be that $p_1 \not\mid q_1 \cdot q_2 \cdots q_k,$ otherwise this would contradict that each $q_j$ is prime and has only itself and $1$ as positive factors. By substituting $q_1 \cdot q_2 \cdots q_k$ for $n,$ $p_1 \not\mid n.$
$p_1 \mid n$ and $p_1 \not\mid n$ is a contradiction, which means the original assumption was false. It must follow that every positive integer's prime factorization is unique up to the ordering of its prime factors.