A funny proof, but what is wrong with it?

[jobsism]

Here's a question:-

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?

 

[micromass]

You've made a logic mistake.

Consider the sentence

"All men are mortal"

This is true. But what's the negation of this sentence?? Is it

"All men are immortal"?

or is it

"There is a man that is immortal"??

It's the second one.

So, if your assertion is

"For all natural numbers n holds the inequality blabla"

then the negation of that is

"There is a natural number n such that the inequality does not hold"

Your proof shows that the inequality holds for n=2. But this does not contradict the negation, as there could still be an n such that it doesn't hold.

 

[bmxicle]

The assumption that for all possible integers n, n!>[(n+1)/2]^n is violated by your counter example, so that demonstrates that your assumption is false. You have not checked all possible values of n so it says nothing about any of those values. Your assumption being false says nothing about the case of only some n's satisfying the inequality.

 

[jobsism]

Oh...now I understand. Thanks, micromass and bmxicle! :D

 

[CellCoree]

I appreciate your curiosity and desire to understand the flaw in this funny proof. The mistake lies in the assumption made in the beginning of the proof. By assuming that [(n+1)/2]^n < n!, we are essentially assuming that the statement we are trying to prove is false. This is known as proof by contradiction, and it is not a valid method of proof in mathematics. In order to prove a statement, we must use logical steps and evidence to show that it is true, not assume that it is false and then prove that this assumption leads to a contradiction. Additionally, the example used to demonstrate this assumption being false (n=2) does not hold true for all positive integers n, so it cannot be used as evidence for the overall statement. Overall, this funny proof may seem convincing at first, but upon closer examination, it is not a valid proof and cannot be relied upon to support the statement.