Discrete Math about asymptotic behavior

False Claim. 2" = 0(1). (1) False Proof. The pmof is by induction on n where the induction hypothesis P(n) is the assertion (1). Base Case: P(0) holds trivially. Inductive Step: We may assume P(n), so there is a constant c > 0 such that 2" 5 c - 1. Therefore 2"*'=2-2"s(2c)-1, n+1 which implies that 2 inductive step. = 0(1). That is, P(n + 1) holds. 1which completes the proof of the We conclude by induction that 2" = 0(1) for all n. That is. the exponential function is bounded by a constant. QED. Identify a major mistake in the above Fake Proof