Induction

Mathematical Induction Example 4 --- Inequality on n Factorial


Problem: For every tex2html_wrap_inline24 , tex2html_wrap_inline26 .

Proof:
In this problem tex2html_wrap_inline28 .
Basis Step: If n = 4, then LHS = 4! = 24, and tex2html_wrap_inline34 .
Hence LHS > RHS .
Induction: Assume that tex2html_wrap_inline26 for an arbitrary tex2html_wrap_inline24 . -- Induction Hypothesis
To prove that this inequality holds for n+1, first try to express LHS for n+1 in terms of LHS for n and try to use the induction hypothesis.
Note here (n + 1)! = (n + 1) n!.
Thus using the induction hypothesis, we get (n + 1)! = tex2html_wrap_inline54 .
Since tex2html_wrap_inline24 , (n+1) > 2.
Hence tex2html_wrap_inline60 .
Hence tex2html_wrap_inline62 .
End of Proof.