[azeemba]

It helped me to look up an example of a space that is not "complete".

Turns out, rational numbers are the classic example of a space that is not complete. A sequence of rational numbers can approximate pi but pi itself doesn't exist in the space (since its irrational). So the rational numbers that get closer and closer to pi form a limit to a value that's not in the space.

[Paul-Craft]

That's a great example. To make it concrete, you can take the sequence such that $a_n$ is the $n$-th partial sum of any of the series here that involve only rational numbers: https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF... multiplied by an appropriate constant.