Originally Posted by

**andrewkoch** Hi everyone! I'm trying to solve a problem related to metric spaces and I'm having a really hard time. Here's the given statement:

Let (A,d) be a metric space of sequences defined as follows:

1) Any sequence in A is a sequence of natural numbers

2) For all (a_n)_n, a_n<a_(n+1) (strictly increasing sequence)

3) For every sequence a in A, the limit as n→∞ of (#{j:a_j<n})/n exists. We call that limit l(a)

Prove that (A,d) is bounded and separable.

Now, if a and b are two sequences in A, d is the distance defined as d(a,b)= k^(-1)+|l(a)-l(b)| where k={min j : a_j≠b_j}. By the way, I've already checked this indeed defines a distance.

The sequence (a_n)_n=(1,2,3,4,...,...) where a_n=n for all n in N is a sequence that exists in A.

I've tried to prove that (A,d) is bounded by assuming A is unbounded and trying to arrive to an absurd. I supposed that there exists a in A such that for all M>0, d(a,x)>M for all x in A. I couldn't get to anything.

For the part of separability I have no idea although I've tried to find a dense countable subset in (A,d). I welcome any suggestions or ideas.