MTH U550 QUIZ 3 Answers




  1. Give the definitions for the following.

    1. $\QTR{cal}{F}$ is a fine family means: For any $\epsilon >0$ there is an interval $I$ in $\QTR{cal}{F}$ with $\ell (I)<\epsilon $.

    2. $A$ is a real number means that $A$ is a fine and consistent family.

    3. The real numbers $A$ and $B$ are equal means that $A\sim B$ as families ($A$ consistent with $B$).

    4. MATH means that $A$ is a real number and $A>0$ or $A<0$.

    5. The family $\QTR{cal}{C}2=\{$rational intervals MATH

  2. State the Wiggle Lemma and give an example of where we used it (you needn't give a whole proof).

    If $x\leq y+\epsilon $ for every $\epsilon >0$ then $x\leq y$. This was used to prove, for example, that if $x$ and $y$ lie in every interval of a fine family, then $x=y$.

  3. Characterize each of the following in terms of lower and upper bounds:

    1. $\QTR{cal}{F<G}$

      There is some upper bound of $\QTR{cal}{F}$ that is less than some lower bound of $\QTR{cal}{G}$.

    2. $\QTR{cal}{F\leq G}$

      Every lower bound of $\QTR{cal}{F}$ is $\leq $ every upper bound of $\QTR{cal}{G}$.

  4. Multiplicative cancellation says that if MATH and $AC=BC$, then $A=B$. Use this principle to prove MATH ($A$ is in MATH of course).

    Let $C=\dfrac{1}{A}$. Then MATH; MATH also. Thus MATH so MATH.