Let $X$ be the set of all real-valued bounded sequences. Let the function $d: X \times X \to [0,∞)$ be $$ d(x, y) := \sup_{i \in \mathbb{N}}\lvert x_i - y_i \rvert $$ where $x = \{x_i\}_{i\in \mathbb{N}}, y = \{y_i\}_{i\in\mathbb{N}} \in X$.
Show that $F = \{x \in X \mid d(x, 0) \leq 1\}$ is a bounded closed set which is not compact. Here $0$ denotes the zero constant sequence.
Hint :
Do you know in a metric space closed ball is always a closed set?
What is the definition of a bounded set? Is the diameter of $F$ finite?
If so what about the sequence $(e_n)_{n\in\Bbb{N}} $ where $e_n$ is sequence $1$ in the $n$-th place and zeros everywhere.
Can a sequence $(x_n) $ have a convergent subsequence sequence where $d(x_m, x_n) =1$ for $m\neq n$ ?
