Found what I was looking for! (my question was a bit far away, I have to admit!): en.wikipedia.org/wiki/Specker_sequence

Perfect! Thanks a lot. Do you have any pointers to those things?

I guess that's it. But I can remember reading a couple of month ago some kind of non convergent sequence on the real line. Could it be that constructivsm reject something at one point, and therefore manage to exhibit such a sequence?