Power to Real Number by Decimal Expansion is Uniquely Defined

Theorem
Let $r \in \R_{>1}$ be a real number greater than $1$, expressed by its decimal expansion:
 * $r = n \cdotp d_1 d_2 d_3 \ldots$

The power $x^r$ of a (strictly) positive real number $x$ defined as:
 * $(1): \quad \ds \lim_{k \mathop \to \infty} x^{\psi_1} \le \xi \le x^{\psi_2}$

where:

is unique.

Proof
If $r$ is rational this has already been established.

Let $d$ denote the difference between $x^{\psi^1}$ and $x^{\psi^2}$:

It follows from Nth Root of 1 plus x not greater than 1 plus x over n that:
 * $d < \dfrac {x^{n + 1} \paren {x - 1} } {10^k}$

Thus as $k \to \infty$, $d \to 0$.

The result follows from the Squeeze Theorem.