show that $(i+k)! > i!k!$ algebraically

Question

I apologize in advance for asking such elementary question but I would like to show that $$(i+k)! > i!k!$$ I believe induction can prove it but is there an algebraic proof on proving about it?

Answer 1

Yes. Rewrite $$(i+k)!=i!\, (i+1)\dotsm(i+k)$$ and observe each $i+j$, for $1\le j\le k$, is $>j$, so $$\prod_{j=1}^k(i+j)>\prod_{j=1}^k j=k!$$

Added, on a suggestion of N. F. Taussig:

We have a strict inequality only if both $i$ and $k$ are non-zero.