What does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers.

(i) $[0,1]$ with the Euclidean metric.

(ii) $\mathbb{Q}$ with the Euclidean metric.

(iii) The subset

$$\{(0,0)\} \cup\{(x, \sin (1 / x)) \mid x>0\} \subset \mathbb{R}^{2}$$

with the metric induced from the Euclidean metric on $\mathbb{R}^{2}$.

Write down a metric on $\mathbb{R}$ with respect to which $\mathbb{R}$ is not complete, justifying your answer.

[You may assume throughout that $\mathbb{R}$ is complete with respect to the Euclidean metric.