Hausdorff spaces & Continuous Mappings via Convergence.

[strangerep]

Homework Statement

A kind person is helping me to self-study some aspects of topology,
continuity, etc. He posed the following exercise for me, which I
can't do, but he doesn't have time to write up the full solution.

Ex: Show that a mapping $$f$$ between Hausdorff spaces is continuous
if and only if the sequence $$f(x_0), f(x_1), \dots$$ converges to $$f(x)$$
for every sequence $x_0, x_1, \dots$ converging to $x$.

Homework Equations

The Attempt at a Solution

To show f is continuous, one must show that every open set in the range of f(x) must
have an inverse image which is also an open set. I also know the definition
of a Hausdorff space as a topological space in which any two points are
contained in disjoint open sets. But that's about as far as I get. Does this
theorem have a name that I could look up? Or can anyone give me some
more hints about how to progress the proof?

TIA.

 

[EnumaElish]

From http://en.wikipedia.org/wiki/Hausdorff_spaces#Properties

Almost all spaces encountered in analysis are Hausdorff; most importantly, the real numbers are a Hausdorff space. More generally, all metric spaces are Hausdorff. In fact, many spaces of use in analysis, such as topological groups and topological manifolds, have the Hausdorff condition explicitly stated in their definitions.

Which tells me you can start from the reals (as a special case) then generalize.