Characterisation of Non-Archimedean Division Ring Norms/Sufficient Condition/Lemma 3
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Theorem
Let $\sequence {x_n}$ be the real sequence defined as $x_n = \paren {n + 1}^{1/n}$, using exponentiation.
Then $\sequence {x_n}$ converges with a limit of $1$.
Proof
We have the definition of the power to a real number:
- $\paren {n + 1}^{1/n} = \map \exp {\dfrac 1 n \map \ln {n + 1} }$
For $n \ge 1$ then $n + 1 \le 2 n$.
Hence:
\(\ds \frac 1 n \map \ln {n + 1}\) | \(\le\) | \(\ds \frac 1 n \map \ln {2 n}\) | Logarithm is Strictly Increasing | |||||||||||
\(\ds \) | \(\) | \(\ds \) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 n \paren {\ln 2 + \ln n}\) | Logarithm on Positive Real Numbers is Group Isomorphism | |||||||||||
\(\ds \) | \(\) | \(\ds \) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\ln 2} n + \frac 1 n \ln n\) |
- $\ds \lim_{n \mathop \to \infty} \frac 1 n \ln n = 0$
By Sequence of Reciprocals is Null Sequence:
- $\ds \lim_{n \mathop \to \infty} \frac 1 n = 0$
By Combined Sum Rule for Real Sequences:
- $\ds \lim_{n \mathop \to \infty} \paren {\frac {\ln 2} n + \frac 1 n \ln n} = \ln 2 \cdot 0 + 0 = 0$
By the Squeeze Theorem for Real Sequences:
- $\ds \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} = 0$
Hence:
- $\ds \lim_{n \mathop \to \infty} \paren {n + 1}^{1/n} = \exp 0 = 1$
and the result follows.
$\blacksquare$