# Prove that metric space is complete/incomplete

I have a function $$\rho(x,y) = |x^{1/3} - y^{1/3}|$$ and I need to prove if the function is metric, and if it is, the next step is to prove if this metric space is complete.

So metric is a function with following conditions:

1. $$\rho(x,y) \iff y=x$$
2. $$\rho(x,y) = \rho(y,x)$$ because of absolute value's properties
3. $$\rho(x,z) \leq \rho(x,y) + \rho(y,z)$$ — I can't prove it myself

But I know that the function is metric indeed and how do I prove that this metric space is complete?

I will assume that your space is $$\mathbb R$$. Let $$d$$ be the usual metric on $$\mathbb R$$. Then$$\begin{array}{ccc}(\mathbb R,\rho)&\longrightarrow&(\mathbb R,d)\\x&\mapsto&\sqrt[3]x\end{array}$$is an isometry. So, since $$(\mathbb R,d)$$ is complete, so is your space.
• This is an isometric homeomorphism, that's why $(\mathbb{R},\rho)$ is complete. – Rodrigo Dias May 10 at 12:50