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I am giving it everything, but i just can't get it right.

The problem: Prove by induction that $n!>2^n$ for all integers $n\ge4$

I know how to solve the basic induction problems, but no matter what I do, I can't get this one right.

I saw how the problem is solved (schoolmate uploaded it on our class dropbox), but I just don't understand it.

The problem is, we only worked on the basics.

If someone could explain this like I'm five, that would be appreciated

mythic
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    I highly doubt a five years old can understand induction..... – N. S. Oct 18 '14 at 14:52
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    What specifically do you not understand? – Brad Oct 18 '14 at 14:52
  • Welcome to Math SE ! Are you sure you want to start using induction ? In any case, congratulations ! – Claude Leibovici Oct 18 '14 at 15:02
  • This is the solved problem by my schoolmate, using the "procedure" we used in class.

    I understand that the basis is only valid for numbers bigger or equal to 4, that's more than logical to me.

    Where I get stuck is the third line. We know that $n!>2^n$ is true, but how is that equal to 1 as seen in line 4?

    Also some basic pointers as to how can I improve my induction problem solving would be great, cause I basically spent the whole day practicing induction and I still have an issue understanding harder problems, like this one.

     – mythic Oct 18 '14 at 15:17

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For $n=4$, we have that $4!=24>16=2^4$, so the condition holds. Now assume it's valid for $k\geq4$: $k!>2^k$. Then $$(k+1)!=(k+1)k!>(k+1)2^k>2\cdot2^k=2^{k+1},$$ and we hake that the condition holds for $k+1$.

Mateus Sampaio
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