Consider the set of sequences of integers

$$X=\left\{\left(x_{1}, x_{2}, \ldots\right) \mid x_{n} \in \mathbb{Z} \text { for all } n\right\}$$

Define

$$n_{\min }\left(\left(x_{n}\right),\left(y_{n}\right)\right)= \begin{cases}\infty & x_{n}=y_{n} \text { for all } n \\ \min \left\{n \mid x_{n} \neq y_{n}\right\} & \text { otherwise }\end{cases}$$

for two sequences $\left(x_{n}\right),\left(y_{n}\right) \in X$. Let

$$d\left(\left(x_{n}\right),\left(y_{n}\right)\right)=2^{-n_{\min }\left(\left(x_{n}\right),\left(y_{n}\right)\right)}$$

where, as usual, we adopt the convention that $2^{-\infty}=0$.

(a) Prove that $d$ defines a metric on $X$.

(b) What does it mean for a metric space to be complete? Prove that $(X, d)$ is complete.

(c) Is $(X, d)$ path connected? Justify your answer.