Suppose currency consists of 3 and 4 cent coins. Suppose you want to buy an item that is worth 9 cents. Show that if you have an unlimited number of 3 and 4 cent coins you can buy anything greater than or equal to 6 cents without receiving change.
*looking for guidance in solving, not a completed proof
Suppose you can pay for $n$ cents without receiving change.
Claim: You can pay for $n+1$ cents without receiving change.
Proof: If the payment method for $n$ cents contains at least one $3$ cents coin, replace it by $4$ cents. If it contains only $4$ cent coins, it contains at least 2 of them (assuming $n \geq 6$). Replace two 4 cent coins by three $3$ cent coins.
For $n = 6$ we can pay with two $3$ cent coins. It now follows from induction that we can pay for all $n \geq 6$ cents without receiving change. Q.E.D