# Power Function on Base Greater than One is Strictly Increasing/Real Number

## Theorem

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

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

- $\map f x = a^x$

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

Then $f$ is strictly increasing.

## Proof

Let $x, y \in \R$ be such that $x < y$.

Let $\delta = \dfrac {y - x} 2$.

From Rational Sequence Decreasing to Real Number, there is some rational sequence $\sequence {x_n}$ that decreases to $x$.

From Rational Sequence Increasing to Real Number, there is some rational sequence $\sequence {y_n}$ that increases to $y$.

From Convergent Real Sequence is Bounded:

- $\exists N_1 \in \N: n \ge N_1 \implies x - \delta < x_n < x + \delta$

Since $\sequence {x_n}$ is decreasing:

- $n \ge N_1 \implies x \le x_n < x + \delta$

From Convergent Real Sequence is Bounded:

- $\exists N_2 \in \N : n \ge N_2 \implies y - \delta < y_n < y + \delta$

Since $\sequence {y_n}$ is increasing:

- $n \ge N_2 \implies y - \delta < y_n \le y$

Let $N = \max \set {N_1, N_2}$.

Then, for $n \ge N$:

\(\displaystyle x\) | \(\le\) | \(\displaystyle x_n\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle x + \delta\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y - \delta\) | |||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle y_n\) | |||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle y\) |

From Power Function on Strictly Positive Base is Continuous and Sequential Continuity is Equivalent to Continuity in the Reals:

- $x_n \to x \implies a^{x_n} \to a^x$
- $y_n \to y \implies a^{y_n} \to a^y$

Also, from Power Function on Base Greater than One is Strictly Increasing: Rational Number:

- $\sequence {a^{x_n} }$ decreases to $a^x$

and:

- $\sequence {a^{y_n} }$ increases to $a^y$.

So, for $n \ge N$:

\(\displaystyle a^x\) | \(\le\) | \(\displaystyle a^{x_n}\) | as $\sequence {a^{x_n} }$ decreases to $a^x$ | ||||||||||

\(\displaystyle \) | \(<\) | \(\displaystyle a^{y_n}\) | Power Function on Base Greater than One is Strictly Increasing: Rational Number | ||||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle a^y\) | as $\sequence {a^{y_n} }$ increases to $a^y$ |

Hence the result.

$\blacksquare$