Prove that a sequence of subsequential limits contains inf and sup

[Sick0Fant]

Okay. The problem I have is:

Let {x_n} be bdd and let E be the set of subsequential limits of {x_n}. Prove that E is bdd and E contains both its lowest upper bound and its greatest lower bound.

So far, I have:
{x_n} is bdd => no subseq of {x_n} can converge outside of {x_n}'s bounds=>E is bounded.
Now, sse that y=sup(E) is not in E=> there is a z in E s.t. y-e < z < y for some e > 0.

Now, how would one proceed from here?

 

[mathman]

You can generate a sequence of z's by using a sequence of e's that goes to 0. This sequence of z's the must converge to y.

 

[Sick0Fant]

I already had thought of that: you have y - e< z < y. Take e to be 1/k with e going to infinity, then {z_k} cgt to y, but what can we really conclude from that? Is there any guarantee that a {z_k} is in the original seq?

 

[mathman]

If you can't find a z for any e>0, then y is > sup(E)