I wanted to verify whether not my attempt at the following proof is okay or not:
Claim: Every natural number n ≥ 18 can be written as a sum of 4s and 7s.
My attempt:
Base cases:
n = 18 = 4 + 7(2)
n = 19 = 4(3) + 7
n = 20 = 4(5)
n = 21 = 7(3)
Induction Hypothesis: Assume that for k ≥ 21, each number from 18...k can be written as a sum of 4s and 7s. For k+1, write it as (k-3) + 4. k-3 does, in fact, fall between 18...k so (k-3) can be written as a sum of 4s and 7s. Now just add 4 to express k + 1 as a sum of 4s and 7s. QED