prove by using induction that every integer k, k>1, can be written as a product of primes,
Use strong induction. Say $\displaystyle k>1$, if it is a prime then proof complete. Otherwise, $\displaystyle k=ab$ where $\displaystyle a,b<k$. But then by strong induction both $\displaystyle a,b$ can be written as a product of primes themselves. Thus, $\displaystyle k$ is a product of primes.