An induction I'm struggling with.
Prove $2^n\cdot n! ≤ (n+1)^n$ by induction.
An idea was to show that $2^n\cdot n! ≤ 1+n^2$ since $1+n^2 ≤ (n+1)^n$ using Bernoulli. However the inequality is just wrong so that approach doesn't work. I had the intuition that $2^n ≤ n!$ but I don't think that yields anything for this problem.
I would really like to get a hint or two. Of course you can post your answer, this is obviously what this platform is for, but I won't read them until I solved the problem myself. It's an induction, can't be that difficult right?