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Guidelines for Writing Proofs

  1. When you write mathematics you are writing for a certain audience. In each assignment the mathematical background of the reader should be clear from the instructions. The background of the reader determines how you write. For example, you would write differently for a beginning calculus student than you would for a more advanced student. Unless told otherwise, assume that you are writing for the other students in Math 3283.

  2. Assume that your writing is going to be published, perhaps in a book. In particular, always WRITE IN COMPLETE SENTENCES. The following type of writing is considered a complete sentence. (See page 46 of the Notes).

    $ \textnormal{ Then for all } x, $

    $\displaystyle \left\vert x^n \right\vert < \varepsilon$

    $\displaystyle \Longleftrightarrow \left\vert x \right\vert ^n < \varepsilon ~~~~ (\textnormal{since } \left\vert x^n \right\vert = \left\vert x \right\vert ^n)$

    $\displaystyle \Longleftrightarrow n \ln \left\vert x \right\vert < \ln \varepsilon$

    % latex2html id marker 410$\displaystyle \Longleftrightarrow n > \frac{\ln \va......left\vert x \right\vert < 0 \textnormal{ since } \left\vert x \right\vert < 1).$

    Similarly, the following is okay (see page 38 of the notes):

    $ \textnormal{ We have, }$

    $\displaystyle x \in B$

    $\displaystyle \Longrightarrow -x \in A$

    $\displaystyle \Longrightarrow a \leq -x ~~~~ (\textnormal{since } a \textnormal{ is a lower bound of } A)$

    $\displaystyle \Longrightarrow x \leq -a.$

    Another example:

    $ \textnormal{ Given } \varepsilon \textnormal{ we must find an } n_0 \textnormal{ such that for all } n \geq n_0, $

    $\displaystyle \left\vert s_nt_n - L_1L_2 \right\vert < \varepsilon.$


    $ \textnormal{ Now using the triangle equality, }$

    $\displaystyle \left\vert s_nt_n - L_1L_2 \right\vert$

    $\displaystyle = \left\vert (s_nt_n - s_nL_2) + (s_nL_2 - L_1L_2) \right\vert$

    $\displaystyle \le \left\vert (s_nt_n - s_nL_2) \right\vert + \left\vert (s_nL_2 - L_1L_2) \right\vert$


    $\displaystyle = \left\vert s_n \right\vert \left\vert t_n - L_2 \right\vert + \left\vert L_2 \right\vert \left\vert s_n - L_1 \right\vert .$

  3. Justify your statements. Remember that the purpose is to convince the reader that what you are saying is true. So you want to give reasons for statements that are not completely obvious. For example, if you are using a previous result or a previous definition, this should be indicated. For example, the above statement Now using the triangle inequality ... justifies (at least one of) the inequalities that follow. Similarly, the above statement

    (since a is a lower bound of A)

    justifies the line on which it appears.

    Which statements that you justify and how you go about it depends on the mathematical background of the reader. What is obvious to you might not be obvious to a beginning calculus student. So in writing for such a beginning student you would have to justify statements that you might not need to justify to a more advanced student.

  4. Give the reader a road map. If you are writing a proof of a mathematical statement, let the reader know at the beginning of the proof what is going to happen. This is especially important if the proof is somewhat long and complicated. But it is often helpful even in simpler situations. Here is an example from a text on mathematical logic.

    Theorem. A set of wffs is satisfiable iff every finite subset is satisfiable.

    Proof. The proof consists of two distinct parts. In the first part we take our given finitely ... In the second part we utilize $\Delta$ To make a ... For the first part let ... This concludes the first part of the proof. For the second part ...

    Notice how the reader is told what is happening, and is also kept informed about what is happening as the proof progresses. This makes it much easier for the reader to understand what is going on.

  5. The assignment should be typed and double or triple spaced. Math symbols may be put in by hand. (If you are unable to deliver a typed version, it should be neatly printed.)

  6. When you have finished an assignment, read it out loud. Pretend that you are hearing it for the first time. Often when you do this you will recognize mistakes. Read it to a friend. Even better, read it to someone who does not like you very much. They may find things wrong with it that you did not.