An induction I'm struggling with.
Prove 2n⋅n!≤(n+1)n by induction.
An idea was to show that 2n⋅n!≤1+n2 since 1+n2≤(n+1)n using Bernoulli. However the inequality is just wrong so that approach doesn't work. I had the intuition that 2n≤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?
Answer
Hint: The induction step goes as follows:
2n+1(n+1)!=2nn!2(n+1)≤(n+1)n2(n+1)=2(n+1)n+1
Thus you are left to prove that 2(n+1)n+1≤(n+2)n+1, which is pretty easy.
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