Prove the following statemnet: any natural number greater than 1 is divisible by a prime number

Let's assume now that k is composite with the following form [tex] k = a_1 b_1[/tex] where [tex]a_1, b_1[/tex] represent integers less than the number k.

Assuming that [tex] a_1[/tex] is prime then we have that [tex] a_1 | k[/tex] and we satisfy the conditions.

Now let's assume that [tex] a_1[/tex] is also composite with the following form [tex] a_1 = a_2 b_2[/tex] as the product of two integers and for [tex] a_2, b_2[/tex] we assume that both are integers >1 and smaller than [tex]a_1[/tex]

If we continue this process t times dividing the composite factors into products of smaller factors we satisfy the condition that the set {[tex] a_1, a_2, a_3,...,[/tex]} would be non empty.

And using the Well ordering principle we have t elements. From this we can conclude that [tex] a_t[/tex] is prime and if we discompose this number we satisfy the existance of [tex] a_{t+1}[/tex] less than [tex] a_t[/tex] because we have this:

[tex] a_t |a_{t-1}|..... |a_!|n[/tex] with k a divisible prime.

And with this we satisfy the conditions and then the proof is complete.