The space $(F,d_\infty)$ is not complete. Find explicitly a nested family of closed balls $(K_n)_{n=1}^\infty$ with $r_n \mapsto 0 $ such that $\cap_{n=1}^\infty K_n = \emptyset$ Here, $F$ is the set of all finite sequences and $d_\infty(x,y) = sup_n |x_n - y_n|$

For the solution I know that for complete spaces $\cap_{n=1}^\infty K_n \ne \emptyset$, and it implies that for an incomplete space $\cap_{n=1}^\infty K_n = \emptyset$ but I do not know how to find closed balls to prove this.

  • 1
    $\begingroup$ Hint: use a Cauchy sequence that does not converge (there are standard examples) and for each point $x_n$ in the sequence choose an appropriate radius $r_n$ such that $r_n\to0$, then take $K_n=B(x_n, r_n) $. $\endgroup$ – Jason Nov 8 '17 at 19:40
  • $\begingroup$ @Jason how do I take closed ball as equal to open ball ? $\endgroup$ – Pumpkin Nov 8 '17 at 19:42
  • $\begingroup$ I am referring to the closed ball. $\endgroup$ – Jason Nov 8 '17 at 19:44

I assume you meant by finite sequence a sequence in $\mathbb R$ or $\mathbb C$ with finite support.

For $n\in \mathbb N$ let $a^n \in F$ be definite by $$ a^n_m = \begin{cases} 2^{-m}, & m \le n \\ 0, & m > n. \end{cases} $$ Further let $r_n = 2^{-n}$ and $$K_n = \{ x\in F \mid d(x, a^n) \le r_n \}. $$

Then, $K_n$ is decreasing and the intersection is empty.


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