For each statement, prove that it is true or show that it is false.
- .
- .
- for every .
- for all .
- True.
Proof. We can compute using the definition of the exponential - False.
On the left we haveWhile on the right we have,
But since , these two quantities cannot be equal.
- True.
Proof. The proof is by induction. For the case on the left we haveWhile on the right we have
Therefore, indeed for the case .
Assume then that the statement is true for some positive integer . Then,
Thus, the inequality holds for the case ; hence, it holds for all positive integers
- False.
From the definitions of and we haveUsing these definitions, the inequality states
However, this is false if since for .