(More like comments than another answer, but I don't have enough reputation to comment.)

(1) It might be interesting, but is not quite explicit in François' answer, that this is undecidable even if the input is given as a Turing machine rather than just an oracle: For any TM $M$, we can construct a TM $M'$ that outputs François' sequence. (Hardwire $M$ into $M'$; then for each $k$, simulate $M$ for $k$ steps on blank input and output the appropriate number.) We still cannot decide if the limit of the sequence is zero by examining $M'$, unless we can decide if $M$ halts.

(2) Also maybe interesting -- I believe that, in computable analysis, the usual model is that a real number $a$ is given as you propose, but on both sides (an infinite sequence of intervals, each contained in the previous one, with limit equal to $a$). So your question is *almost* whether equality on the reals is decidable (except that the second number, $b$, is rational, but this doesn't seem easier); and undecidability of equality on the reals is a primary result in computable analysis!