Power Function on Base Greater than One is Strictly Increasing/Integer

Theorem
Let $a \in \R$.

Let $a > 1$.

Let $f : \Z \to \R$ be the real-valued function defined as:
 * $f \left({ k }\right) = a^k$

where $a^k$ denotes $a$ to the power of $k$.

Then $f$ is strictly increasing.

Proof
By Exponential with Base Greater than One is Strictly Increasing/Natural Number, the theorem is already proven for positive integers.

Let $i, j$ be integers such that $i < j < 0$.

Hence the result.