Prove that "2^n\\geq2n, \\forall n\\geq1"
For any integer n ≥ 1, let "P_n" be the statement that
Base Case. The statement "P_1" says that
which is true.
Inductive Step. Fix "k\\geq1," and suppose that "P_k" holds, that is,
It remains to show that "P_{k+1}" holds, that is,
"2k\\geq2, \\forall k\\geq1""2^k\\geq2k=>2\\cdot2^k\\geq2\\cdot2k=>""=>2^{k+1}\\geq4k=2k+2k\\geq2k+2=2(k+1)"
Therefore "P_{k+1}" holds.
Thus, by the principle of mathematical induction, for all "n\\geq1,P_n" holds.
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