Now for the solution to the question in my previous post, which
asked what you can learn about , given the sequence of integers
. Nick Johnson commented: Well, the obvious thing one can learn
given just |(10^n)r| is the first n digits of the decimal
expansion of r. For example, when we are given , we can say with
confidence that . If we wait long enough, we can learn as many
decimal digits of as we want; in particular, to learn the first
n decimal digits of , we can just wait for the th item of the
list. This seems like an awfully long time to wait, though. Not
only that, but if we only look at the first, tenth, hundredth,
thousandth,… elements of the sequence, we would be ignoring
almost all the information in the sequence! Intuitively, it
seems that we should be able to do better by taking more of the
sequence into account. And indeed, we can. The key is to realize
that Do you see why? Taking the floor of something never makes
it bigger, so (in fact, it can never be equal since is
irrational, but that’s not