Limits of sequences
We spend some time considering different flavors of “limit behavior”: stabilization, approximate stabilization etc. A sequence is called converging, if it -stabilizes for any positive accuracy .
To show that passing to a limit “respects” arithmetic operations, we need to work out a bit of “interval arithmetic” with a special attention to the division which may cause problems.
We discuss the weaker notion of a partial limit (accumulation point) and study under what assumptions a unique partial limit is the genuine limit.
Finally, we show that monotone bounded sequences always converge. This is one of the most powerful tools to show that the limit exists when it is not possible to compute it explicitly.