Chapter 1

Suppose \(\left\\{a_{i}\right\\}_{i \in I}\) and \(\left\\{b_{i}\right\\}_{i \in J}\) are sequences in \(\mathbb{Q}\) with $$\lim _{i \rightarrow \infty} a_{i}=\lim _{i \rightarrow \infty} b_{i}$$ Show that \(a_{i} \sim b_{i}\).

Show that the relation \(R\) on \(\mathbb{Z}\) defined by \(m \sim_{R} n\) if \(m\) divides \(n\) is reflexive and transitive, but not symmetric.

Suppose \(\left\\{a_{i}\right\\}_{i \in I}\) is a Cauchy sequence which is not bounded away from \(0 .\) Show that the sequence converges and \(\lim _{i \rightarrow \infty} a_{i}=0 .\)

Show that for any \(a \in \mathbb{Q}\), one and only one of the following must hold: (a) \(a<0,\) (b) \(a=0,(\mathrm{c}) a>0\).

Show that the relation \(R\) on \(\mathbb{Z}\) defined by \(m \sim_{R} n\) if \(m-n\) is even is an equivalence relation.

Suppose \(\left\\{a_{i}\right\\}_{i \in I}\) is a Cauchy sequence which is bounded away from 0 and \(a_{i} \sim b_{i} .\) Show that \(\left\\{b_{j}\right\\}_{j \in J}\) is also bounded away from \(0 .\)

Show that if \(a, b \in \mathbb{Q}^{+}\), then \(a+b \in \mathbb{Q}^{+}\).

Given an equivalence relation \(R\) on a set \(A\), show that a. \([x] \cap[y] \neq \emptyset\) if and only if \(x \sim R y ;\) b. \([x]=[y]\) if and only if \(x \sim_{R} y\).

Show that if \(u \in \mathbb{R},\) then one and only one of the following is true: (a) \(u>0,\) (b) \(u<0,\) or (c) \(u=0\)

Suppose \(a, b, c \in \mathbb{Q}\). Show each of the following: a. One, and only one, of the following must hold: (i) \(ab\). b. If \(a0\) and \(b>0,\) then \(a b>0\).

Show that if \(a, b \in \mathbb{R}^{+}\), then \(a+b \in \mathbb{R}^{+}\).

Show that if \(a, b \in \mathbb{Q}\) with \(a>0\) and \(b<0\), then \(a b<0\).

Show that \(\mathbb{R}\) is an ordered field, that is, verify the following: a. For any \(a, b \in \mathbb{R},\) one and only one of the following must hold: (i) \(ab\) b. If \(a, b, c \in \mathbb{R}\) with \(a0\) and \(b>0,\) then \(a b>0\).

Show that if \(a, b, c \in \mathbb{Q}\) with \(a0\) and \(a c>b c\) if \(c<0\)

Show that if \(a, b \in \mathbb{R}\) with \(a>0\) and \(b<0\), then \(a b<0\).

Show that if \(a, b \in \mathbb{Q}\) with \(a

Show that if \(a, b, c \in \mathbb{R}\) with \(a0\) and \(a c>b c\) if \(c<0\)

Show that for any \(a \in \mathbb{Q},-|a| \leq a \leq|a| .\)

Show that if \(a, b \in \mathbb{R}\) with \(a

Show that the supremum of a set \(A \subset \mathbb{Q},\) if it exists, is unique, and thus justify the use of the definite article in the previous definition.

Show that for any \(a \in \mathbb{R},-|a| \leq a \leq|a|\).

Show that there does not exist a rational number \(s\) with the property that \(s^{2}=3\).

Show that if \(A \subset \mathbb{R}\) is nonempty and has a lower bound, then inf \(A\) exists. (Hint: You may wish to first show that inf \(A=-\sup (-A)\), where \(-A=\\{x:-x \in A\\})\).

Show that there does not exist a rational number \(s\) with the property that \(s^{2}=6\)

Let \(A=\left\\{a: a \in \mathbb{Q}, a^{3}<2\right\\}\). 1\. Show that if \(a \in A\) and \(ba\), then \(b \notin A\).