Let me give you a few examples:

To make a sequence with subsequential limits 0 and 1, how about:

0, 1, 0, 1, 0, 1, 0, 1, ....

or

1/3, 2/3, 1/4, 3/4, 1/5, 4/5, ..., 1/n, (1-1/n), ...

To make a sequence with subsequential limits 0 and 1 and -1, how about:

, which is 0, 1, 0, -1, 0, 1, 0, -1, ...

To make a sequence with subsequential limits e and infinity, how about:

for even and for n odd.

For the subsequential limit set being countable, I'd suggest trying to make it the positive integers - something easy.

Can you blend the following infinite set of sequences into a *single* sequence?

1, 1, 1, 1, ...

2, 2, 2, 2, ...

3, 3, 3, 3, ...

4, 4, 4, 4, ...

etc.

(Hint: do you recall the method that's usually used to prove that a countable union of countable sets is a countable set?)