# Harrison Chapman

## Math 317: Homework 2

Due: Friday, September 13, 2019
1. For each subset of $\mathbb R$ below, determine both the supremum and the infimum. You do not need to give a rigorous proof of your answer.

1. Supremum 5, infimum 1

2. Supremum 1/2, infimum -1

3. Supremum 1, infimum -1

4. Supremum $$\sqrt 5$$, infimum $$-\sqrt 5$$

5. Supremum 1, infimum 1/2

2. Let $A$ be a nonempty, bounded set of real numbers. Let $-A$ be the set $\{ -x \;:\; x \in A \}$. Prove that

Let $$M = \sup A$$, which exists and is real as $$A$$ is bounded. So $$\forall a \in A, M \ge a$$. Then $$\forall y=-a \in -A$$, $$M \ge a \Rightarrow -M \le -a = y$$. So $$-M$$ is a lower bound for $$-A$$.

Suppose $$L$$ is any lower bound for $$-A$$. So $$\forall a \in A$$, $$L \le -a \Leftrightarrow -L \ge a$$ so $$-L$$ is an upper bound for $$A$$, and in particular $$-L \ge M$$ as $$M$$ is the least upper bound of $$A$$, and we see that $$L \le -M$$, so $$-M$$ is the greatest lower bound for $$-A$$.

That is, $$\inf(-A) = -\sup A \Leftrightarrow \sup A = -\inf(-A).$$

3. Give examples of;

1. A sequence of irrational numbers converging to a rational number

For instance, $$(s_n) = (\pi/n)$$.

2. A sequence of rational numbers converging to an irrational number

For instance, $$(s_n) = ((1+\frac 1n)^n)$$.

4. For each of the following sequences, determine its limit (usual Calculus reasoning will be helpful here), then prove that the sequence does indeed converge to this limit.

1. Let $\epsilon > 0$. Consider $N = \frac 19 \left( \frac {17}{\epsilon} - 12 \right)$. Then for $n > N$,

So $\lim a_n = 2/3$.

2. Let $\epsilon > 0$. Consider $N = 1/\epsilon$ and $n > N$. Then

So $\lim (\cos n)/n = 0$.

5. Let $(s_n)$ be a sequence in $\mathbb R$.

1. Prove that $\lim{s_n} = 0$ if and only if $\lim{ s_n^2 } = 0$. (Hint: This is an “if and only if” statement, so you will need to prove both directions of the statement.)

($\Rightarrow$) Suppose $\lim s_n = 0$. Consider $\epsilon > 0$. Then there exists $N$ so that $n > N$ implies that $% $.

So then $% $. So $\lim s_n^2 = 0$.

($\Leftarrow$) The reverse direction is very similar to the forward direction.

2. Prove that if $s_n = (-1)^n$ that $\lim{s_n^2}$ exists but $s_n$ diverges.

We have that $s_n$ diverges by Example 4 in Section 8 (pp41-42). Also, $s_n^2 = 1$ which converges to $1$ as it is constant.