Prove that there exists $i, j \in \mathbb{N}$ such that $n=3i+5j$ for $n\ge 8$
I'm having a hard time with this exercise, I'm trying to prove it by induction:
Basis step:
$n=8 \implies 8=3\cdot1+5\cdot 1$
$n=9 \implies 9=3\cdot3+5\cdot0$
$n=10 \implies 10=3\cdot0+5\cdot2$
Induction step:
If it's true for $n=h$ then it must be true for $n=h+1$.
So now, I don't know how to begin proving that $k+1=3i+5j$.