Probability Models proof question

I have two questions.. One is for my first assignment of the year, and the other one is just a general probability query.

First: the assignment question. In our notes, a proof is given for:

A1 ⊆ A2 ⊆ ... = P(lim[n→∞]An) = lim[n→∞]P(An)

I have a feeling that the proof is the same as its opposite, except that the lim is approaching 0 instead of ∞. We are asked to show this proof, if anyone could correct me I'd appreciate it!

Also, my lecturer has been using Def 1.1, 1.2 etc and Lemma 1.1 a) b) Lemma 1.2 a) - e) etc. Is this a known way to announce the properties of probability models, or is she just making it easier for us?

Cheers (by they way, this is my first post on the site, so please take it easy on me! (Worried))