/sci/ - Science & Math

>>7122863

I like xkcd and smbc.
Also abstruse goose

2 Cauchy sequences are going out tonight, they find a "no-limit" party in town and try to enter the nightclub, but the bouncer says "sorry, we're complete".

>>6479579

>>6368836
abstruse goose is funny

>>6339901

>>6326400
notta gif but I couldn't resist

So I am working through Munkre's Topology and think I have come up with a negative proof. Can you all confirm or point out my flaw?
Here is the problem:
Let <span class="math">I_{n} \subset R,~ n \epsilon N[/spoiler] be a sequence of closed intervals so that <span class="math">I_{k+1} \subset I_{k}[/spoiler]. Prove that <span class="math"> \cap_{k \epsilon N} I_{k}[/spoiler] is non-empty.

My proof-negative: (some of the first few things might be non- sequiturs)
Assume <span class="math"> \cap I_{k} = \emptyset[/spoiler]
By the statement of the problem, and by definition of subset, <span class="math"> I_{k+1} \subset I_{k}[/spoiler], which implies <span class="math"> \forall x \epsilon I_{k+1}, x \epsilon I_{k}[/spoiler]. For <span class="math"> \cap I_{k} [/spoiler] to be empty, there must exist a <span class="math">I_{k}= \emptyset[/spoiler]. The empty set is both open and closed, so <span class="math"> \cap I_{k}[/spoiler] can be empty iff <span class="math"> \exists I_{k}= \emptyset[/spoiler].


And hopefully this TeX works.