completeness of the metric $[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$

Question

I must determine if the following space is complete

$[1,2]\times[1,2]\rightarrow \mathbb R:$ $d(x,y)=|x^4-y^4|$.

Is $([1,2],d) $complete?

my try:

Let $\{x_n\}$ be a Cauchy sequence

by definition

$\forall \epsilon, \exists n_0, \forall n,m \geq n_0 $ $d(x_n,x_m)\leq \epsilon$

$4|x_n-x_m| \leq|x_n^4- x_m^4|\leq256|x_n-x_m|$

then I don't know how to continue Any help will be much appreciated

Answer 1

Here's one way that uses the inequality you just found. We have $$ 4|x_n - x_m| \leq |x_n^4 - x_m^4| \leq \epsilon.$$ So $(x_n)_{n \geq 1}$ is Cauchy in the usual metric. Note $\mathbb{R}$ is complete with its usual metric. In fact $[1, 2]$ is complete with its usual metric as well. (Why?) Can you reason from here why your space is complete?