If we allow the metric to be $d(x,y)=|x-y|$, we must prove that this is complete. Now, I have proven all properties of a metric space. However, I don't particularly now where to begin to prove that this is complete. I understand that we need every Cauchy sequence in this space to converge insid...

Problem Which one of the following metric is not complete on $\mathbb R$: $|x-y|$ $|arctan(x) - arctan(y)|$ $|x^3 - y^3|$ $|x^{1/3} - y^{1/3}|$ 1 is complete. I am not sure about the 2, 3 and 4. For 2, I do not know the definition of $arc$ function For $3. 4$, they can be viewed as the spe...

How to know if these two spaces are complete or not ? $(\mathbb{R}, d)$ where $d(x,y)=|x^3-y^3|$ $(\mathbb{R}, d')$ where $d'(x,y)= \ln(1+|x-y|)$ thank you very much

I have to check for completeness of following metric spaces $1)$ : $\mathbb{R}$ with metric defined by $d_1(x,y) =\mid e^x - e^y \mid$ for all $x, y \in \mathbb{R}$. $2)$: $\mathbb{Q}$ with metric defined by $d_2(x,y) = 1$, for all $x, y\in \mathbb{Q}, ~~ x \neq y$. I have shown that $(\mat...

Please ,how to prove that the space $\mathbb{R}$ endowed with the metric $d(x,y)=|e^x-e^y|$ is not a complete space? I don't find a Cauchy sequence but not convergent Please Thank you.