# Proof

There are four cases, how the real number $$b$$ can influence the convergence of the real sequence $$(b^n)_{n\in\mathbb N}$$:

### Case $$(1)$$: $$|b| < 1$$

In this case, the sequence $$(b^n)_{n\in\mathbb N}$$ is convergent. This follows immediately from the corollary to the Archimedean Axiom about the existence of arbitrarilly small powers.

### Case $$(2)$$. $$b = 1$$

In this cases, it follows trivially that $$\lim_{n\to\infty} b^n=1$$.

### Case $$(3)$$: $$b = - 1$$

Because the set $$(\mathbb R, + ,\cdot)$$ of real numbers is a field, the set $$(\mathbb R\setminus\{0\},\cdot)$$ is an Abelian group. Therefore, it follows from the definition of the exponentiation for Abelian groups that

$b^n := \begin{cases} 1 & \text{ if } n=0 \\ b\cdot b^{n-1} & \text{ if } n > 0 \end{cases}$

It can be proven that $$-(-b)=b$$ for all $$b\in\mathbb R$$. Therefore, also $$-(-1)=(-1)(-1)=(-1)^2=1$$. It follows that the real sequence $$(-1)^n_{n\in\mathbb N}$$ is alternating between the values $$1$$ and $$-1$$. It is simply the sequence $$(1,-1,1,-1,1,-1,\ldots)$$. Since this sequence neither converges to the number $$1$$, nor it does converge to the number $$-1$$, it is divergent, by definition.

### Case $$(4)$$: $$|b| > 1$$

In this case, the sequence $$(b^n)_{n\in\mathbb N}$$ tends to infinity. This follows immediately from the corollary to the Archimedean Axiom about the existence of powers exceeding positive real numbers.

Github: ### References

#### Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983