# Proof? (Real Analysis 1)

1. Aug 28, 2010

### phillyolly

1. The problem statement, all variables and given/known data

See attachment

2. Relevant equations

3. 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.

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2. Aug 28, 2010

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 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 $$x \in A \rightarrow x \in B.$$