NOTE: ``Prove'' means give a careful, well-explained
1. Let be a sequence of real numbers,
and suppose that the two numbers L and L' satisfy
Prove from the definition of limit that L = L'.
2. a) Give an example of a sequence which
has no convergent subsequences.
b) Give an example of a sequence which
does not have a limit, but which has a subsequence which does
have a limit. c) Give an example of a bounded set of real numbers which does
not contain its least upper bound.
3. The Tetrabonacci numbers are defined by , and
for . Suppose that the limit exists and is equal
to L. Calculate L.