I have spent a considerable amount of time attempting to derive the inequality 8^sqrt(7) < 7^sqrt(8) and have gotten tantalizingly close, but not completed the thing.
I have seen a few "proofs," but have not understood them in their entirety, particularly:
"It suffices to prove that
8^sqrt(7) < 7^[31 sqrt(7) / 29]
8 / 7 < 7^30 / 8^28."
How on Earth do these two inequalities relate? I would appreciate a brief explanation and/or some references. I do admit that inequalities are a weak spot where I am concerned. I have no real "feel" for them. I tried to correct that this year when I developed several of them for a small project in Real Analysis. Still, I am not comfortable with them.
Wait. . . nevermind. <Grin> I believe it goes like this: