Hi I submitted this as a graded assignment and received a poor grade. Could someone help me see what was wrong with my proof.
Let n be a nonnegative integer. Show that 2n2≥n!
Proof
(i) Base Case
For n = 0
We have 202≥0!
Which Yields, 1≥1
Thus the base case holds.
(ii) Inductive Hypothesis:
Assume for some k∈Z,k≥0 that ,2k2≥k! then look at k+1
2(k+1)2=2k2+2k+1=2k2⋅22k⋅2≥k!⋅22k⋅2 via inductive hypothesis
We now take k!⋅22k⋅2 and relate it to (k+1)!
k!⋅22k⋅2≥(k+1)!k!⋅22k⋅2≥(k+1)⋅k!22k+1≥(k+1)
Thus the statement holds for k+1
Therefore by the generalized principle of mathematical induction,
2n2≥n! for n∈Z,n≥0
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