choose the complete metric space?

Question

let $d(x,y)$ be the usual Euclidean metric on $\mathbb{R}^2$.

which of the following metric spaces is complete ?

1. $\mathbb{Q}^2 \subset \mathbb{R}^2$ with the metric $d(x,y)$

2. $(0,\infty) \times [0,\infty) \subset \mathbb{R}^2$ with the metric $d(x,y)$

3. $[0,1] \times [0,1) \subset \mathbb{R}^2$ with the metric $d''(x,y) = \min \{1,d(x,y)\}$.

4. none of these

my attempts ; i know that $\mathbb{Q}$ is not a complete metric space so $\mathbb{Q}^2$ is also not a complete metric space ..so option $1$ is not true...and for option $2$

....$d(x,y)$ may or may not be complete metric space. suppose i take

$d(x,y)= |e^x - e^y|$ which is not complete

now the for option $3$.. $d''(x,y) = \min\{1,d(x,y)\} = 1$ is complete space

as every Cauchy sequence is constant so it is convergent. so the correct

answer is option $3$ that $d''(x,y) = \min\{1,d(x,y)\}$ is complete metric space.

is my answer is correct or not ...pliz verified and tell me the solution . i would be more thankful

Answer 1

A subspace of a complete metric space is complete if and only of it is a closed subset of the whole space. Therefore, 1. and 2. are false.

The third option is also false. The sequence $\left(0,1-\frac1n\right)_{n\in\mathbb N}$ is a Cauchy sequence which is not convergent.

Therefore, the correct option is 4.