Defining Sequence of Natural Logarithm is Convergent
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Theorem
Let $x \in \R$ be a real number such that $x > 0$.
Let $\left\langle{ f_n }\right\rangle$ be the sequence of mappings $f_n : \R_{>0} \to \R$ defined as:
- $f_n \left({ x }\right) = n \left({ \sqrt[n]{ x } - 1 }\right)$
Then $\left\langle{ f_n }\right\rangle$ is pointwise convergent.
Proof
Fix $x \in \R_{>0}$.
From Defining Sequence of Natural Logarithm is Strictly Decreasing, $\left\langle{ f_n \left({ x }\right) }\right\rangle$ is strictly decreasing.
From Lower Bound of Natural Logarithm/Proof 3, $\left\langle{ f_n \left({ x }\right) }\right\rangle$ is bounded below.
From Monotone Convergence Theorem, $\left\langle{ f_n \left({ x }\right) }\right\rangle$ is convergent
Hence the result, by definition of pointwise convergence.
$\blacksquare$