This question is based on Goldbach's Proof of the Infinitude of Primes

Goldbach's Proof of the Infinitude of Primes (1730)
Let

a1 = a

a2 = a1+b

a3 = a1a2 + b

a4 = a1a2a3 + b

..

..

Prove that the above seq is paiwise prime (given HCF(a,b) = 1)

My attempt

If the assertion is not true then their exist a smallest number 'n' for which a_n is not relatively with some a_m (m>n)

Thus there exist a prime p such that

p|a_m

and p|a_n

=> p|a_m - a_n

=> p|a1a2a3....a_(n-1) [a_n....a_(m-1) - 1]

Now p can't divide [a_n....a_(m-1) - 1] (as p|a_n)

Thus p|a1a2a3....a_(n-1)

=> p|ai where i < n

Hence 'n' is not the smallest. Hence Proved

I have two questions

1. Can someone please review this proof for me?

2. Can someone suggest a proof without using induction?