So in a previous proof I had to show that given $y\in\mathbb{R},n\in\mathbb{N}$ and $\epsilon >0$, show that for some $\delta>0$, if $u\in\mathbb{R}$ and $|u-y|<\delta$ then $|u^n-y^n|<\epsilon$. After some trouble I was able to prove this but now once again I am having trouble with this statement.
The current proof I'm working on that given $x > 0$ and $n\in\mathbb{N}$, prove that there is a unique $y > 0$ such that $y^n = x.$ The problem says I am supposed to do this using the previously mentioned proof, however other than the fact that both have a power of $n$ in statement I can't see the relation.
I know that I should probably use l.u.b. and the fact that the previous proof showed that $f(y)=y^n$ is continuous but I'm not sure where to start putting the pieces together.