Frobenius coin problem induction help I need some help with the actual Induction proof of a Frobenius coin problem. This is the exact problem: 
The government of Elbonia has decided to issue currency only in 5 and 9 cent denominations. Prove that there is largest value that Elbonians cannot pay with this denomination
And a later question says to prove that all values over this found value is payable with 5 and 9 cent coins.
First of all, I've found the largest value that can't be paid is 31 cents. I've found this by just writing out the combinations of coins until I got 5 in a row, and then each of those numbers can just have 5 added to them to continue forever, starting at 32. However I'm not sure if that could be considered "proof" and if I need to show this in a more official way.
I've started on the induction proof anyway, but I'm having some trouble as where to go. I know to prove this I need to show that for S(n): where n is the amount payable with 5 or 9 cent pieces, show that 
S(n) -> S(n+5),
S(n+1) -> S(n+6),
S(n+2) -> S(n+7),
S(n+3) -> S(n+8),
S(n+4)-> S(n+9),
So for my base case I've let n=32, and shown that 
32 = 3(9) + 3(5),
33= 2(9) + 3(5),
34 = 1(9) + 5(5),
35 = 7(5),
36 = 4(9)
So I've shown my base cases can be paid with 5 and 9 cent pieces, but now I'm stuck. What exactly do I assume for my inductive assumption? That S(k), s(k+1).. etc is true for some kEZ? Normally when we are given induction questions for the inductive step there is a way to rearrange it to make your assumption show up somewhere to help you prove it, but I can't see how to do that for the inductive step here.
Any help on this would be awesome, sorry for the long question! Thanks!

You are all set. From your base cases (32 to 36) you can add 5 to each to get the next stretch of 5 (37 to 41), and from them the next one (42 to 46), and...
To make it formal:
Bases: As you did show, 32 to 36 are all possible.
Induction: Asuming all between $32$ and $n \ge 32$ are representable,
$n + 1$ is representable. If $n \le 36$, $n + 1$ is part of the base. Otherwise it is just adding a 5 coin to $n + 1 - 5 = n - 4 > 32$, which is representable by induction hypothesis.
