- #1
Sick0Fant
- 13
- 0
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?
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?