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The finite interval assertion in the last sentence is clearly false, since you can take an infinite subsequence of rationals with nested neighborhoods. I don't see a way to rescue the argument (but it's more or less moot at this point).
I should mention that there is a difference between TMF (the 576-periodic spectrum) and tmf (the connective cover). TMF(point) maps to modular forms with possibly negative weight, that may have poles at the cusp, while tmf(point) maps to traditional modular forms.
