# Proof

(related to Proposition: General Powers of Positive Numbers)

We have already proven that rational exponents, the power function is identical to the exponential function of general base, i.e. $a^x=\exp_a(x),\quad\quad x\in\mathbb Q.$ Now, we will prove that the expression $$a^x$$ makes perfectly sense and this equation also holds1, if we allow $$x$$ to be a general real number. According to the definition of real numbers, a real number $$x$$ is a class of all rational Cauchy sequences, which equal each other except a difference, which is a rational Cauchy sequence converging to zero, formally $x\in\mathbb R\Longleftrightarrow x=(x_n)_{n\in\mathbb N}+I$ where $$(x_n)_{n\in\mathbb N}$$ is a rational Cauchy sequence representing the real number $$x$$, and

$I:=\{(i_n)_{n\in\mathbb N}~|~i_n\in\mathbb Q,\lim i_n=0\}$ is the set of all rational sequences, which converge to $$0$$. Now, we have

$\begin{array}{rcll} a^x&=&\lim_{n\to\infty}a^{x_n+i_n}&\text{due to the definition of real numbers}\\ &=&\lim_{n\to\infty}\exp_a(x_n+i_n)&\text{definition of rational powers of positive numbers}\\ &=&\lim_{n\to\infty}\exp_a(x_n)\cdot \exp_a(i_n)&\text{functional equation of exponential function of general base}\\ &=&\exp_a(x)\cdot \exp_a(0)&\text{continuity of exponential function of general base}\\ &=&\exp_a(x)\cdot \exp(0\cdot \ln(a))&\text{definition of exponential function of general base}\\ &=&\exp_a(x)\cdot \exp(0)&\text{multiplication by }0\\ &=&\exp_a(x)\cdot 1&\text{proposition stating that} \exp(0)=1\\ &=&\exp_a(x)&\text{multiplication by }1\\ \end{array}$

Above, we have used, among others, the following propositions: * functional equation of exponential function of general base, * continuity of exponential function of general base, and the * proposition stating that $$\exp(0)=1$$.

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### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
2. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013

#### Footnotes

1. E.g. a general power $$3^\pi$$ makes perfectly sense, although it makes no sense to say that we "multiply $$\pi$$ copies of $$3$$ together".