Is the following Proof Correct ? I am required to construct the proof using Strong Induction
Theorem. The Martian monetary system uses colored beads instead of coins. A blue bead is worth $3$ Martian credits, and a red bead is worth $7$ Martian credits. Then for all $n\ge 12$, there is some combination of blue and red beads that is worth $n$ credits.
Proof. For the case $n\in\{12,13,14\}$ the following combinations $$\begin{cases} 12 = 3\cdot4 +7\cdot 0\\ 13 = 3\cdot6 +7\cdot 1\\ 14 = 3\cdot0 +7\cdot 2\\ \end{cases}$$ are sufficient. Now assume for an arbitrary $n\in\{15,16,17,...\}$ that for all $k\in\{12,13,14,...,n-1\}$ there exists integers $a$ and $b$ such that $k = 3\cdot a+7\cdot b$. Evidently $n = (n-3)+3$ since $n\ge 15$ it follows that $m = n-3\ge 12$ consequently using inductive hypothesis we have integers $x$ and $y$ such that $m = 3\cdot x+7\cdot y$ thus $n = m+3 = 3\cdot (x+1)+7\cdot y$
$\blacksquare$