Defining Sequence of Natural Logarithm is Convergent

From ProofWiki
Jump to navigation Jump to search

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.

$\blacksquare$