What is the relationship between first countable spaces and Hausdorff spaces?

[Pyroadept]

Homework Statement

Let X be a first countable space where no sequence has more than one limit. Show that X must be Hausdorff.

Homework Equations

The Attempt at a Solution

Hi everyone,
Here's what I've done so far:

I used this thm: If X is a Hausdorff space, then sequences in X can have at most one limit. (but not necessarily the converse)

So X is potentially a Hausdorff space as all it's sequences have only one limit, if at all.

But then I'm completely stuck as to where to go from here. Obviously it's got something to do with being first countable, but I can't see what! Can anyone please give me a point in the right direction?

Thanks for any help!

 

[micromass]

Assume that our space is not Hausdorff. Take x and y points where the Hausdorffness space (thus every neighbourhood of x intersects every neighbourhood of y)
Let the neighbourhood base of x be {B1,B2,...}, were $$...\subseteq B_3\subseteq B_2\subseteq B_1$$.
Likewise, let the neighbourhood base of y be {C1,C2,...}, were again $$...\subseteq C_3\subseteq C_2\subseteq C_1$$.

Try to construct a sequence that converges that converges to both x and y.