Q:

Every integer greater than 1 is divisible by a prime

Every integer greater than $1$ is divisible by a prime. 
Prove it by mathematical induction (of weak form).

A:

Let $P(n)$ be the assertion that every integer $i$ such that $2\le i\le n$ is divisible by some prime, that is, there is a prime that divides $i$. We prove by weak induction that $P(n)$ holds for every integer $n\ge 2$. That will imply in particular that every integer $n\ge 2$ is divisible by some prime.
The weak induction proof: It is clear that $P(2)$ is true. Now suppose that $P(k)$ is true for some given $k$. We show that $P(k+1)$ is true. 
Certainly every $i$ with $2\le i\le k$ is divisible by some prime, by the induction assumption. So we need only prove that $k+1$ is divisible by some prime.
If $k+1$ is prime then we are finished. If not, there exist integers $a$ and $b$, with $2\le a\lt k+1$, and $2\le b\lt k+1$, such that $k+1=ab$. By the induction assumption, since $2\le a\le k$, the number $a$ is divisible by some prime. It follows that $k+1$ is, and we are finished.
Remark: Note that the argument is less natural than the strong induction argument. The "trick" was to change the induction assumption. 

