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Proposition: Limit of Nth Root of a Positive Constant

Let $(a_n)_{n\in\mathbb N}$ be a real sequence defined by $a_n=\sqrt[n]{a}$ for some positive real number $a > 0$ and all $n\in\mathbb N$, where $\sqrt[n]{a}$ denotes its nth-root. Then limit of $(a_n)_{n\in\mathbb N}$ exists and we have $$\lim_{n\to\infty} a_n=\lim_{n\to\infty}\sqrt[n]{a}=1,\quad\quad (a > 0).$$

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References

Bibliography

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