$\forall a\in \mathbb R^+ \space\forall n \in \mathbb N \space\space\exists x\in \mathbb R^+ \space\space x^n=a$ I'm trying to prove a theorem. This theorem somehow states that Real Numbers have roots of every degree like $d \in \mathbb N$.  By $\mathbb R^+$, I mean the Real numbers which are greater than zero.  
Theorem :
$\forall a\in \mathbb R^+ \space\forall n \in \mathbb N \space\space\exists x\in \mathbb R^+ \space\space x^n=a$  
Question :
Prove the theorem written above.  
Note : I've seen some related examples like proving that there exists a real number like $x$ that $x^3=5$. In those examples, i knew what $a$ is. So i could conclude that a set like $A=\{x\in\mathbb R:x^3\le5\}$ has a smallest upper bound like $S$. Then i'd try to prove that $S=x$. But in this general form, I can't use the same method.  

Consider (for $a>0$) $$x:=e^{\frac 1n \ln(a)}.$$
