The lower bound on a non-degenerate American-style put (that is, a put where there is still some

Question:

The lower bound on a non-degenerate American-style put (that is, a put where there is still some uncertainty about whether S T > X or not) is:

Assume that ST = 0 and rtT = 0. Common sense says that you should exercise the put, since the exchange rate cannot fall any further. Yet the bound Pt > X says that the put should trade above its intrinsic value. Where is the fallacy?

Exchange Rate
The value of one currency for the purpose of conversion to another. Exchange Rate means on any day, for purposes of determining the Dollar Equivalent of any currency other than Dollars, the rate at which such currency may be exchanged into Dollars...