Real positive number/Root/Unique existence/Fact/Proof

Proof

We define recursively nested intervals ${}[a_{n},b_{n}]$. We set

${}a_{0}=0\,$

and we take for ${}b_{0}$ an arbitrary real number with ${}b_{0}^{k}\geq c$. Suppose that the interval bounds are defined up to index ${}n$, the intervals fulfil the containment condition and that

${}a_{n}^{k}\leq c\leq b_{n}^{k}\,$

holds. We set

${}a_{n+1}:={\begin{cases}{\frac {a_{n}+b_{n}}{2}},{\text{ if }}{\left({\frac {a_{n}+b_{n}}{2}}\right)}^{k}\leq c\,,\\a_{n}{\text{ else}},\end{cases}}\,$

and

${}b_{n+1}:={\begin{cases}{\frac {a_{n}+b_{n}}{2}},{\text{ if }}{\left({\frac {a_{n}+b_{n}}{2}}\right)}^{k}>c\,,\\b_{n}{\text{ else}}.\end{cases}}\,$

Hence one bound remains and one bound is replaced by the arithmetic mean of the bounds of the previous interval. In particular, the stated properties hold for all intervals and we have a sequence of nested intervals. Let ${}x$ denote the real number defined by this nested intervals according to fact. Because of exercise, we have

${}x=\lim _{n\rightarrow \infty }a_{n}=\lim _{n\rightarrow \infty }b_{n}\,.$

Due to fact  (2), we get

${}x^{k}=\lim _{n\rightarrow \infty }a_{n}^{k}=\lim _{n\rightarrow \infty }b_{n}^{k}\,.$

Because of the construction of the interval bounds and due to fact, this is ${}\leq c$ but also ${}\geq c$, hence ${}x^{k}=c$.

To fact