For each statement, prove that it is true or show that it is false.
- for every .
- for all .
Proof. We can compute using the definition of the exponential
On the left we have
While on the right we have,
But since , these two quantities cannot be equal.
Proof. The proof is by induction. For the case on the left we have
While 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
From the definitions of and we have
Using these definitions, the inequality states
However, this is false if since for .