These problems are part of the homework due on Monday, July 7. (Note: I made a minor correction to #2 a couple hours after the initial posting.)

1. For each integer $n\ge 1$, let $A_n$ be the set defined by $$A_n = \{x\in\mathbb R ~|~ n^2 \le x \le (n+10)^2\}.$$ Determine the sets $\bigcup\limits_{n=1}^\infty A_n$ and $\bigcap\limits_{n=1}^\infty A_n$. Prove your answers.
2. Let $f:\mathbb R-\{-1\}\to\mathbb R$ be defined by $f(x)=\dfrac x{x+1}$. Determine the range of $f$ and prove your answer.
3. Let $F:\mathbb R\times\mathbb R\to\mathbb R\times\mathbb R$ be defined by $F(x,y)=(x+y,x-y)$. Show that $F$ is a bijection, and determine its inverse function.
4. Let $S$ be the set of infinite sequences of integers. For example, the entire Fibonacci sequence $$(1,1,2,3,5,8,13,\ldots)$$ is one element of $S$. Show that $S$ is not countable. (Hint: Use an argument similar to the proof that $\mathbb R$ is uncountable.)