Can you check if my proof is right?
Theorem. ∀x≥8,x can be represented by 5a+3b where a,b∈N.
Base case(s): x=8=3⋅1+5⋅1✓x=9=3⋅3+5⋅0✓x=10=3⋅0+5⋅2✓
Basis stays true, because 8,9,10∈S
Lets assume that x∈S. That means x∈an∨x∈bn∨x∈cn.
If x∈an then x+1∈bx,
If x∈bx then x+1∈cx,
If x∈cx then x+1∈ax.
I can’t prove that but it’s obvious. What do you think about this?
Proof by induction.
For the base case n=8 we have 8=5+3.
Suppose that the statement holds for k where k>8. We show that it holds for k+1.
There are two cases.
1) k has a 5 as a summand in its representation.
2) k has no 5 as a summand in its representation.
For case 1, we delete “that 5” in the sum representation of k and replace it by two “3“s ! This proves the statement for k+1.
For case 2, since k>8, then k has at least three “3“s in its sum representation. We remove these three 3‘s and replace them by two fives! We obtain a sum representation for k+1. This completes the proof.