my homework do it

Prove by induction that each positive integer n ≥ 8 can be represented in the form
n = 3a + 5b, where a and b are non-negative integers.

>>3
At step 3, you forgot "and (c - a) an element of N", in other words, c >= a. This is why the conclusion in step 5 is wrong - you can't express n=1 with the given constraints.

Real answer:
n = {8, 9, 10} can be solved by hand. "n >= 11 is expressible" follows from n-3 being expressible.