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Homework Statement
Show that n!>2n for all n>3.
Homework Equations
I will attempt to use induction.
The Attempt at a Solution
We want to show that n!>2n for all n>3.
Consider the case when n=4.
[tex] 4! = 24 > 2^4 =16.[/tex]
We want to show by way of induction that if the inequality is true for some k greater than 4, it is true for k+1.
Assume the inequality holds for k>4. Then,
[tex] 2^k < k! [/tex]
[tex] 2*2^k < 2k![/tex]
[tex]2^{k+1} < 2k![/tex]
for k>4.
But 2k! < (k+1)! for all k>4. Therefore
[tex]2^{k+1} < (k+1)![/tex]
and so by induction it follows that n! > 2n for all n > 3.