I was looking at a math book when I stumbled across a problem:
Prove the following: given I am a little puzzled. How would I prove it? Using proof by counterexample?
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You can prove this using derivatives...
Okay... Is it possible without derivatives?
Why would you want a different method?
First note that for all , while for all . So for all .
Then note that and .
For , we have
So that means grows slower than does, so can never catch up.
Therefore for all .
Okay, that makes sense... So we can use growth rates of functions to determine which is the largest in the end.
Use a contradiction: suppose not, that means that would imply but this is obviously false since
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