"The Principle of Shrinkage"

[Edwin Clark]

Perhaps some of the readers of this list may have a suggestion
concerning a recent question on sci.math.research. The question
arose in a discussion of G. Spenser-Brown's purported recent
proof of the Riemann Hypothesis. In the proof GSB used
a new axiom ("the principle of shrinkage") which Tim Chow
formulated as follows:

  If f:R -> R or f:N -> R is a "naturally occurring function" and
  lim(x->oo) f(x) appears to exist, then it does.

Then Tim asks the question:

Can anyone give an example of a conjecture of the form "lim(x->oo) f(x)
exists" for which there was extensive numerical evidence and which
remained open for some time, but was eventually shown to be false?

For the entire thread that this comes from see
http://groups.google.com/group/sci.math.research/browse_frm/thread/1eee03282e96b657