1549901369-Elements_of_Real_Analysis__Denlinger_

To the Instructor xx vii

But it causes the student to recognize the importance of the hypotheses,

and to rethink what they actually say. Sometimes alternative forms of the

hypotheses are more useful than the original forms.

• Proofs are written, using proper English sentence structure, grammar, and
  punctuation. I feel we must be responsible to our students for modeling
  proper communication style.


• Even though the proofs are in paragraph form, reasons for each "step" are
  given as often as practical. Here, judgment is exercised to avoid choking
  a proof with unnecessary detail.


• The end of a proof is clearly indicated with the customary symbol, •·
  (We frequently use the symbol 0 to denote the end of an example or
  thought, to set it off from the beginning of a new thought.)


• The sophistication of proofs escalates as the book progresses. Proofs later
  in the book tend to be more streamlined than those coming earlier.

USE OF LOGICAL SYMBOLS

Formal logical symbolism makes its first appearance in Chapter 1 and is used
consistently after that. The justifications for this are historical, pedagogical,
and practical- stemming from calculus' need to make use of the concept of
"infinity." Calculus as we know it could not exist without effectively harness-
ing this concept. Faced with the task of making the concept respectable, real
analysts of the past two centuries came up with a brilliant two-step solution.
First, nineteenth century mathematicians discovered how to treat "infinity"
with complete precision using only the tools of finite mathematics: by using
inequalities and logical quantifiers. Then, by developing symbols for the logi-
cal operations and rules for their use, twentieth century mathematicians made
it possible to invoke the relevant logical principles systematically and clearly,
without the "fog" that often beclouds the "words only" approach. My own in-
tellectual development as an undergraduate was profoundly affected by my first
encounter with elementary symboli c logic. I was overwhelmed by its power and
economy. My eyes were opened to how mathematical ideas can be expressed
and how proofs flow.
Thus, you can understand why I resort to logical symbols in putting across
the concepts of analysis. In particular, I find it beneficial to use symbols for the
two logical connectives, :::;. and ¢=>, and the two quantifiers, V and 3. A correct
understanding of "and," "or," and DeMorgan's laws is also very helpful, but we
do not need to use their symbols. Quantifiers are ubiquitous in analysis, and it
is unthinkable to try to learn the subject without coming to grips with them
in some way. For example, the definition of limit involves three quantifiers: V,
3, and a second V. I often leave the third quantifier unstated in order to give
the reader a break, but its presence must be recognized when negating that