Have you calculated the first few $\displaystyle A_i$? The first problem only requires the first four and you should be able to see what others are like.
With i= 1, $\displaystyle A_i$ is the set of all x such that $\displaystyle 1- \frac{1}{1}< x\le 1$ which is the same as $\displaystyle 0< x \le 1$. In interval notation that is (0, 1].
With i= 2, $\displaystyle A_2$ is the set of all x so that $\displaystyle 1- \frac{1}{2}< x\le 2$. In interval notation, that is (1/2, 2].
With i= 3, $\displaystyle A_3$ is the set of all x such that $\displaystyle 1- \frac{1}{3}< x\le 3$. In interval notation, that is (2/3, 3].
With i= 4, $\displaystyle A_4$ is the set of all x such that $\displaystyle 1- \frac{1}{4}< x\le 4$. In interval notation, that is (3/4, 4].
The union of those is the interval (0, 4].
For the last, to prove that two sets, X and Y, are equal, first prove $\displaystyle X\subset Y$, then prove $\displaystyle Y\subset X$. And to prove that $\displaystyle X\subset Y$, start "if $\displaystyle x\in X$ and use the definitions and properties to conclude "therefore $\displaystyle x\in Y$".
For example, given that $\displaystyle A\subset B$, to show that $\displaystyle A\cup B= B$:
First show [b]B\subset A\cup B[/tex]. That's easy. If $\displaystyle x\in B$ then, by definition of "union", $\displaystyle x\in A\cup B$.
Now show [b]A\cup B\subset B[/tex]. If $\displaystyle x\in A\cup B$ then either (1) $\displaystyle x\in A$ or (2)$\displaystyle x]in B$.
1) if $\displaystyle x\in A$ then since we are given that $\displaystyle A\subset B$, $\displaystyle x\in B$.
2) if $\displaystyle x\in B$, we are done.
That makes a lot of sense, thank you so much! I've yet to have a course in discrete math, but ran across these problems so I had no idea where to begin, but was very curious to see the solution. Now I see how obvious it was. Thanks again!