If you're able to find a rapidly converging sequence of approximations, then in finite time you can approximate arbitrarily well. Given the realities of finite numerical precision, this can be exactly as useful as an exact solution.
Sadly, "finite time" is not what we need - they are looking for approximations that will be much rougher than the imprecision of finite numerical precision, and they want the approximation algorithm to have a good provable run time.
In general, taking theoretical approximation algorithms and shrinking the approximation factor to below some threshold where you "tell the difference" is a dead end. For instance, this algorithm's running time has a poly^{1/\epsilon} factor, so you pay a huge cost for it. It is better to think \epsilon = 1/3 or so.
We'll know more tomorrow.