Fundamental theorem of arithmetic. Proof:- $P(n)$: $n$ is a prime or can written uniquely as a product of primes. $P(2)$ is true, and assume true for $n=3,4,...,k$. Now if $k+1$ is not a prime it can be written as $k+1= r\times s$ . Now $r<k+1$ and $s<k+1$. Thus both can be written uniquely as product of primes (or prime themselves). Thus $k+1$ can be written uniquely as product of primes. So $P(n)$ is true for all natural numbers.
Where did I go wrong? Can I include 'uniquely' in the statement? Because in every book it is proved first that every composite number is product of primes and then uniqueness is proved.