Weierstrass... thing

There is in my maths text-book this property/theorem given under the name of Weierstrass property/theorem:

Let $ (a_n) $ be a sequence of real numbers.

a)If $ (a_n) $ is monotonically increasing and has an upper bound, then $ (a_n) $ is convergent.
b)If $ (a_n) $ is monotonically decreasing and has a lower bound, then $ (a_n) $ is convergent.

When searching on Google, I can't find anything related to this, so my first question is:
What is the name of this thing?

What I want to do is to attempt to prove this property/theorem, I would like some hints on what to start with and what should I end with.

Thanks!