- #1

- 109

- 0

Prove that for all positive integers n,

[(n+1)/2]^n >= n!

And here's a funny proof for it:-

Assume to the contrary that, for all positive integers n,

[(n+1)/2]^n < n!

However, for n=2,

(3/2)^2 > 2!

Therefore, our assumption must be false.

And hence, for all positive integers n, [(n+1)/2]^n >= n!

Now, I know that this proof can't be correct, because I've seen the real proof, and it's a marvel, making use of algebraic inequalities, and the proof above simply seems too simple compared to it. But I wonder, what's the mistake with the above proof?