I am so confused with Cauchy Sequences. I was sick the day we learned about them. Here are two problems that I am absolutely unsure about. (just two of many)

1. Prove straight from the definition that is NOT a Cauchy Sequences

and

2. Let be a Cauchy Sequence such that is an integer for every . Show that is ultimately constant.

As I said, I am so lost. Please help? Thanks so much.