Homework 3: Absolute Value, More Supremums

Instructions. Assignments should be stapled and written clearly and legibly.

1. Section 3.2, #6(b), 6(c). (For problem 6(c), I recommend that you first solve problem 4 below.)
2. Let \(a \lt b\) be real numbers and consider the set \(T= \mathbb{Q}\cap [a,b]\). Prove that \(\ssup T=b\). You may use any of the theorems from class.
3. Suppose that \(A\) and \(B\) are nonempty bounded sets in \(\bR\) and that \(\ssup A\lt\ssup B\). Prove that there exists \(b\in B\) that is an upper bound for \(A\). Then show by example that this is not always the case if we only assume \(\ssup A\leq \ssup B\).
4. Let \(a, b\) be real numbers. Prove: If \(a\lt b+\epsilon\) for every \(\epsilon\in \bR\) such that \(\epsilon>0\), then \(a\leq b\).
5. For each of the following, find all real numbers \(x\) which satisfy the inequality. Express your answers using interval notation.
   1. \(|x + 3| \lt 1\)
   2. \(|x-1| + |x-2| \gt 1\)
   3. \( |x-1| + |x+1| \gt 1\)
   4. \( |x-1| - |x+1| \gt 1\)
6. Prove that if \(a\lt x \lt b\) and \(a \lt y \lt b\) then \(|x- y| \lt b-a\).