Good exam problems.

Consider and .1.) Continuous function and also a Cauchy sequence such that is not a Cauchy sequence.

Impossible because continous functions on closed finite intervals are uniformly continous. But the property of uniformly continous functions is that their images of Cauchy sequences still remain Cauchy sequences.2.) Continuous function and also a Cauchy sequence such that is not a Cauchy sequence.

Impossible. Let be Cauchy sequence in . Since it is Cauchy it is bounded. So the entire sequence is contained in where is its upper bound. But since is uniformly continous on it means (as in 2) that is Cauchy.3.) Continuous function and also a Cauchy sequence such that is not a Cauchy sequence.

Draw a picture. Consider on .4.) Continuous function that's bounded on and that has a maximum value on the open interval, however, does not have a minimum value.