- #1

transmini

- 81

- 1

## Homework Statement

Consider the space ##([0, 1], d_1)## where ##d_1(x, y) = |x-y|##. Show that there exists a sequence ##(x_n)## in ##X## such that for every ##x \epsilon [0, 1]## there exists a subsequence ##(x_{n_k})## such that ##\lim{k\to\infty}\space x_{n_k} = x##.

## Homework Equations

N/A

## The Attempt at a Solution

We had no idea where to even begin with this one. It doesn't seem to make sense in the first place when you take the theorem:

Let ##(x_n)## be sequence, and let ##x \epsilon X##. Prove that if ##\lim{n\to\infty}\space x_n = x##, then for every subsequence ##(x_{n_k})## we have ##\lim{k\to\infty} \space x_{n_k} = x##

With the problem saying that there is specific sequence such that for any limit point we can find a subsequence, this implies there a multiple subsequences with different limits, which contradicts what this previous theorem says.

Any suggestions on where to begin with this one?