does every infinite hausdorff space contains a countable infinite discrete subspace?
In a more general light:
folklore theorem: Every infinite topological space contains a homeomorphic copy of one (or more) of the following 5 spaces:
As each of the spaces has the property that every infinite subspace of it is homeomorphic to the whole space, this list is minimal.
And spaces 1-4 are not Hausdorff, which implies what you need, as being Hausdorff is hereditary.
The nicest proof of this I know uses Ramsey's theorem (off hand I do not know a reference, who does?) using a partition of the triples or pairs of X, IIRC.
Reference to the original paper: Minimal Infinite Topological Spaces, John Ginsburg and Bill Sands, The American Mathematical Monthly Vol. 86, No. 7 (Aug. - Sep., 1979), pp. 574-576. The Ramsey proof I saw somewhere else, though.
Not sure whether this is "research-level", but: first show that any infinite Hausdorff space has a proper infinite closed subset. Then proceed by induction.