Lesson 53.

All the usual guidelines apply, and papers will not be graded if these guidelines are egregiously flouted.

Due Friday, April 25:
(I may hand this out in class.)

Last modified Apr 26, 2008 11:48 PM

Section 16:
  1. Definition of Limit.

    Suppose you are given a sequence  (sn)  and a number  L.  Refer to the definition of  “ the sequence  (sn)  converges to  L ”  in the textbook (Sect. 16, Definition 16.2). [Note that the name of the limit of the sequence has been changed from  s  in the textbook to  L  here.]

    1. Rewrite this definition without words, using only mathematical symbols.

    2. Imagine explaining to a student just starting Math 323 how you would go about proving that a sequence  (sn)  converges to a number  L,  using the ideas you have learned in Math 323, following these guidelines:
      1. The definition is a universally quantified statement. Given a sequence  (sn)  and a number  L,  explain precisely how you would BEGIN the proof that this definition is satisfied.
      2. The universal quantification which begins the definition is followed by an existentially quantified statement. Explain the general approach to proving such a statement. (In contrast to question (a) above, you are not asked here to give a precise statement, but rather to describe strategy and tactics for figuring out how to prove the existentially quantified statement, and then what in general you do to give the precise proof.)
      3. The existentially quantified statement just discussed contains another universally quantified statement (about a natural number  n).  Explain precisely how you would BEGIN the proof that this universally quantified statement is true.

  2. Equivalent Definitions of Limit.
    (Notation modified April 26; mathematical content unchanged.)

    Prove that the following two definitions of convergence are equivalent to each other and also to the “original” definition in the textbook. Note the universal quantifiers in boldface! See further instructions below the definitions.

    Definition 1.  sn  converges to  L   means that, for every real number  η > 0,  we can find a (positive) real number  P  such that
    (***(P, η ))        for every natural number  n ≥ P,  |sn - L| < η.

    Definition 2.  sn  converges to  L   means that, for every real number  k > 0,  we can find a natural number  M  such that
    (****(M, k))        for every natural number  n > M,  |sn - L| < k.

    Give detailed proofs, showing that you know how to prove universally quantified statements. Note that each definition contains TWO universal quanitifiers.
    DO NOT try to twist one definition into another. Each definition is a universally quantified statement. Start the proof in the usually way for starting the proof of a universally quantified statement. Each definition contains another universally quantified statement. Start the proof of THIS in the usual way for proving a universally quantified statement. (Smell the clover in Part I.  Avoid the error so many students made in Projects IV and IV ', where they correctly described, in Part I, how to do certain proofs, and then, when doing Part II, they did not follow the “ rules” they had set down in part I !)

    Note that you are supposed to prove that THREE statements are equivalent:  The definition given in the book and the two definitions given here. In doing this, keep in mind that “equivalence” of statements is an equivalence relation!
    Thus, to prove, e.g., that statements  p, q, and r are equivalent, one can prove that  p implies q,  q implies r, and  r implies p,  and to do this efficiently often requires you to THINK about what to choose for  p, q, and r.


Last modified Apr 26, 2008 11:48 PM

Go to Lesson 53a (due Apr 30).

Go to Lesson 54 (due Apr 28).

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