This is an interactive website to help you learn how to turn a proof from the text or from your notes into a proof of a homework problem. Here is a proof I did in class.

### Prove that $$\sup(A\cup B)=\sup\{\sup A, \sup B\}$$.

Proof: Let $$x\in A\cup B$$, and let $$u=\sup\{\sup A, \sup B\}$$. These immediately imply that $$\sup A \le u$$ and $$\sup B \le u$$ because $$u$$ is an upper bound for the two-element set $$\{\sup A, \sup B\}$$. We want to show that $$u$$ satisfies the definition of $$\sup (A\cup B)$$. Because $$x\in A\cup B$$, either $$x\in A$$ or $$x\in B$$, so either $$x\le \sup A\le u$$ or $$x\le \sup B\le u$$. In either case, $$x\le u$$ so $$u$$ is an upper bound for $$A\cup B$$. Now let $$v$$ be any upper bound for $$A\cup B$$. If $$a\in A$$, then $$a\in A\cup B$$ so $$a\le v$$, making $$v$$ an upper bound for $$A$$, from which it follows that $$\sup A \le v$$. Similarly, $$\sup B \le v$$ which implies that $$v$$ is an upper bound for the set $$\{\sup A, \sup B\}$$. Therefore $$u\le v$$ as desired.

How can we adapt this proof to the homework problem I assigned?

### Prove that $$\inf(A\cup B)=\inf\{\inf A, \inf B\}$$.

Try it yourself first, then click the box below to see the solution.

Show the Proof

Proof: Let $$x\in A\cup B$$, and let $$w=\inf\{\inf A, \inf B\}$$. These immediately imply that $$\inf A \ge w$$ and $$\inf B \ge w$$ because $$w$$ is a lower bound for the two-element set $$\{\inf A, \inf B\}$$. We want to show that $$w$$ satisfies the definition of $$\inf (A\cup B)$$. Because $$x\in A\cup B$$, either $$x\in A$$ or $$x\in B$$, so either $$x\ge \inf A\ge w$$ or $$x\ge \inf B\ge w$$. In either case, $$x\ge w$$ so $$w$$ is an lower bound for $$A\cup B$$. Now let $$t$$ be any lower bound for $$A\cup B$$. If $$a\in A$$, then $$a\in A\cup B$$ so $$a\ge t$$, making $$t$$ a lower bound for $$A$$, from which it follows that $$\inf A \ge t$$. Similarly, $$\inf B \ge t$$ which implies that $$t$$ is a lower bound for the set $$\{\inf A, \inf B\}$$. Therefore, $$w\ge t$$ as desired.

The expressions in red needed to be changed, but the rest of the proof stayed the same!