# Decimal Fraction

**1. Decimal Fractions .**

(a) Prove that every rational number has either a terminating or repeating expansion. (Hint. long division.)

(b) In light of the last question, give me a clear way to
generate an

infinite decimal expansion that does not repeat, thus constructing

an irrational number . Be sure I can tell how to generate each digit

and also make an argument why it does not repeat.

(c) Consider an infinite decimal expansion
Write it as

a limit of a sequence of truncated ("cut-off") decimal representations .

(d) Prove that the sequence in the last question is
Cauchy. (Reminder.

a sequence a_{n} is Cauchy if for any , there is
a natural number

N such that if m, n > N, then )

(e) (don't turn in) Remind yourself what a completion of a
metric

space is, and convince yourself that the real numbers are the completion of the rational numbers .

(f) Write 0.999 . . . as the limit of a sequence of
terminating decimals .

Prove that the limit of this sequence is 1. Yes, you'll have to use

an argument.

**2. Comparing Set Cardinalities.** Prove using the
definition of "same

size" and "fits in" in class that the following pairs of sets have the same

size.

(a) Prime numbers and whole numbers.

(b) Terminating decimals and repeating decimals (here you can count

an infinitely repeating 0 as repeating).

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