So it’s easy to show when a=1,b=0 then $a+bsqrt{5} = 1$. So 1 is in S (I assume $0 in mathbb{N}$).

However, I am having difficult with the inductive step. This is my thinking thus far.

Assume $a + bsqrt{5} in S$.

We need to show $(a+1) + bsqrt{5} in S$. Since we know $1+0sqrt{5} = 1 in S$. We can substitute in for 1 as follows: $(a + (1 + 0sqrt{5})) + bsqrt{5} = (a+1) + bsqrt{5}$. Thus, $(a+1) + bsqrt{5} in S$.

Is this correct? If so, why is the substitution I made allowed? I’m new to induction so my apologies if this seems like a trivial question, but that is where I am struggling to be confident in my proof. I feel like I am using what I need to prove to show what I need to prove.

Thank you!