Uniqueness of Positive Root of Positive Real Number/Positive Exponent/Proof 2

Theorem

Let $x \in \R$ be a real number such that $x > 0$.

Let $n \in \Z$ be an integer such that $n > 0$.

Then there is at most one $y \in \R: y \ge 0$ such that $y^n = x$.

Proof

We have that:

$0 < y_1 < y_2 \implies y_1^n < y_2^n$

so there exists at most one $y \in \R: y \ge 0$ such that $y^n = x$.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Real Numbers: $1.37$. Theorem