So, here's my take on both forms of the paradox - Zeno's original, of Achilles and the Tortoise (converging on a distance), and Thomson's variation, the Lamp (converging on a time). In both of them, assuming that the runner / lamp-switcher in question CAN infinitely continue to cover ever smaller increments of distance / time (which, as the article points out, modern physics dictates that they CAN'T), they will never reach the point of convergence.
So, using the example figures from the article: assuming that they both continue along the race-track infinitely, neither Achilles nor the Tortoise will ever reach the 11.111111...m mark. I.e. the real point, in my opinion - apart from Zeno's conclusion that Achilles can never overtake the Tortoise - the real point is that neither of them will even finish the race (or to be more precise, assuming that the race is 100m long, neither of them will finish more than 11.11111...% of the race). They're "infinitely recursing" in their movement, as they approach but never reach the point where one would be a tied winner with the other.
Similarly: assuming that someone continues switching it infinitely, the "state of the lamp" will never reach the 2min mark. I.e. apart from Thomson's conclusion that we can't say whether the lamp will be on or off after 2 mins, the real point is that it's impossible for "the lamp as something that's on or off" to even EXIST when time is gte 2 mins. The lamp will continue "infinitely recursing" in its temporal state, as it approaches but never reaches the point where that dude with a tired index finger could stop and take a break.
In conclusion: Zeno's Paradox / Thomson's Lamp are solvable, if we assume that the universe doesn't exist. :P