Proving Equivalence Statements in Real Analysis 1

[phillyolly]

Homework Statement

See attachment

Homework Equations

The Attempt at a Solution

It is not a homework. I am just reviewing for myself.
This is the very first, basic problem of the first chapter of Real Analysis 1 by Bartle and Sherbert. Proofs of Real Analysis don't make any sense to me.

 

[Raskolnikov]

The "if and only if" denotes an equivalence relation between the two statements, i.e., we have $$(A \subseteq B) \longleftrightarrow (A \cap B = A).$$ There are two steps necessary to proving an equivalence statement:

1. Show $$(A \subseteq B) \rightarrow (A \cap B = A).$$
2. Show $$(A \cap B = A) \rightarrow (A \subseteq B).$$

For both parts, we begin by assuming the respective premise.
For 1, we need to show that $$x \in (A \cap B) \rightarrow x \in A$$ and that $$x \in A \rightarrow x \in (A \cap B).$$
For 2, we just need to show that $$x \in A \rightarrow x \in B.$$
Do all this, and you have yourself a proof!