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.

  • $\begingroup$ This is an isometric homeomorphism, that's why $(\mathbb{R},\rho)$ is complete. $\endgroup$ – Rodrigo Dias May 10 at 12:50
  • $\begingroup$ And there's no need to do anything with Cauchy sequences at all? $\endgroup$ – Alexander May 10 at 12:51
  • $\begingroup$ @RodrigoDias I've edited my answer. Thank you a lot. $\endgroup$ – José Carlos Santos May 10 at 13:02
  • $\begingroup$ @Alexander Only if you really want to. $\endgroup$ – José Carlos Santos May 10 at 13:02

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