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Proposition: Product of a Convergent Real Sequence and a Real Sequence Tending to Infinity
Let $(a_n)_{n\in\mathbb N}$ be a real sequence tending to infinity (i.e. either $+\infty$ or $-\infty$). Let $(b_n)_{n\in\mathbb N}$ be a real sequence tending to some real number $b$, i.e. with $\lim_{n\to\infty} b_n=b.$ Then the real sequence $(b_n\cdot a_n)_{n\in\mathbb N}$ is tending to infinity as follows:
- to $+\infty$, if $(a_n)_{n\in\mathbb N}$ is tending to $+\infty$ and $b > 0$,
- to $+\infty$, if $(a_n)_{n\in\mathbb N}$ is tending to $-\infty$ and $b < 0$,
- to $-\infty$, if $(a_n)_{n\in\mathbb N}$ is tending to $+\infty$ and $b < 0$, and
- to $-\infty$, if $(a_n)_{n\in\mathbb N}$ is tending to $-\infty$ and $b > 0$.
Table of Contents
Proofs: 1
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References
Bibliography
- Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983
