I'm having trouble wrapping my head around proofs by induction with this problem:
Prove by careful induction on $k$ that, if $p$ is a prime number dividing the product $m_1m_2...m_k$ of (ordinary) integers $m_1,...,m_k$ where $k\geqslant1$, then $p$ divides at least one of the integers $m_i, i =1,...,k.$
Trying again based on feedback
So I start with the base case, that is:
$k = 1$
If $p$ divides $m_1m_2...m_k$, that gives us $1/p$, meaning $p|1$ which supports the hypothesis.
So the base case holds.
Now for the Inductive step, we have $k +1$ which would equate to $2$
Then we have $m_1m_2...m_k = 1*2$ and then $1*2/p$.
Since $p∣ab \implies p∣a$ or $p∣b$, we know that $p|1$ or $p|2$.
So the inductive step holds, and so does the hypothesis.