Prove that $(1/n^n)\leq(1/n!)$ for every $n\geq1$

Question

I want to know if my answer is correct:

1) For n=1: $ 1=1$ Correct!
2) Let n=k is an inductive assumption which is correct: $$\frac{1}{k^k}\leq \frac{1}{k!}$$ 3) For n=k+1, we should prove that: $$\frac{1}{(k+1)^{k+1}}\leq \frac{1}{(k+1)!}$$ So, $$\frac{1}{(k+1)!}= \frac{1}{(k+1)k!}\geq \frac{1}{(k+1)k^k}\geq \frac{1}{(k+1)(k+1)^k}= \frac{1}{(k+1)^{k+1}}$$

It's correct also for $n=k+1$, so the inequality $1/(k+1)^{k+1}\leq1/(k+1)!$ is correct for every number $n\geq1$

Answer 1

Your answer is correct! I see nothing wrong with it as far as I know about mathematical induction. Keep up the good work!