# Power Function on Base between Zero and One is Strictly Decreasing/Integer

## Theorem

Let $a \in \R$ be a real number such that $0 < a < 1$.

Let $f: \Z \to \R$ be the real-valued function defined as:

$\map f k = a^k$

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

Then $f$ is strictly decreasing.

## Proof

Let $0 < a < 1$.

By Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer, the theorem is already proven for positive integers.

It remains to be proven over the negative integers.

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

$0 < -j < -i$

So:

 $\displaystyle a^{-j}$ $>$ $\displaystyle a^{-i}$ Power Function on Base between Zero and One is Strictly Decreasing: Positive Integer $\displaystyle \leadsto \ \$ $\displaystyle \frac 1 {a^j}$ $>$ $\displaystyle \frac 1 {a^i}$ Real Number to Negative Power: Positive Integer $\displaystyle \leadsto \ \$ $\displaystyle a^i$ $>$ $\displaystyle a^j$ Ordering of Reciprocals

Hence the result.

$\blacksquare$