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Theorem : If $a\geq 0$ is a nonnegative real number and $n$ is a positive integer, there exists a unique real number $r\geq 0$ such that $r^{n}=a$ .

$$\text{proof 1}$$ Consider the set : $$S:=\{s\in \mathbb{R}:s\geq 0 ,s^n\leq a\}$$

Observe that this is a nonempty set since $0∈S$. and $S$ is also bounded . and $$.\\.\\.\\$$


$$\text{proof 2}$$ We define :

$$E= \lbrace t\in R^{+}\ |\ t^n < a\rbrace $$

$$.\\.\\.\\$$


I want another proof . please help me .thank you .

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1 Answer 1

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By the intermediate value theorem, the continuous strictly increasing function $f:[0,\infty)\to [0,\infty)$ given by $f(x)=x^n$ must be bijective because $\lim_{x\to\infty} f(x)=\infty$.

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