Divisibility induction proof: $8\mid 7^n+3^n-2$

Note that by hypothesis $$8|3^n+7^n-2$$ then $8|3^{n+1}+7^{n+1}-2$ if and only if also $$8|(3^{n+1}+7^{n+1}-2)-(3^{n}+7^{n}-2)=3^n(3-1)+7^n(7-1)=\underbrace{2\cdot 3^n+2\cdot 7^n-4}_{\mathrm{divisible\ by\ 8}}+\underbrace{4+4\cdot 7^n}_{\mathrm{divisible\ by\ 8}}.$$