Baby Rudin Problems

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 19 March 2014

Chapter 3

  1. Prove that convergence of {sn} implies convergence of {|sn|}. Is the converse true?
  2. Calculate limn(n2+n-n).
  3. If s1=2, and sn+1= 2+sn (n=1,2,3,), prove that {sn} converges, and that sn<2 for n=1,2,3,.
  4. Find the upper and lower limits of the sequence {sn} defined by s1=0; s2m= s2m-12; s2m+1=12+ s2m.
  5. For any two real sequences {an},{bn}, prove that lim supn (an+bn) lim supnan+ lim supnbn, provided the sum on the right is not of the form -.
  6. Investigate the behavior (convergence or divergence) of an if
    1. an=n+1-n;
    2. an=n+1-nn;
    3. an=(nn-1)n;
    4. an=11+zn, for complex values of z.
  7. Prove that the convergence of an implies the convergence of ann, if an0.
  8. If an converges, and if {bn} is monotonic and bounded, prove that anbn converges.
  9. Find the radius of convergence of each of the following power series:
    1. n3zn,
    2. 2nn!zn,
    3. 2nn2zn,
    4. n33nzn.
  10. Suppose that the coefficients of the power series anzn are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1.
  11. Suppose an>0, sn=a1++an, and an diverges.
    1. Prove that an1+an diverges.
    2. Prove that aN+1sN+1 ++ aN+ksN+k 1-sNsN+k and deduce that ansn diverges.
    3. Prove that ansn2 1sn-1- 1sn and deduce that ansn2 converges.
    4. What can be said about an1+nan and an1+n2an?
  12. Suppose an>0 and an converges. Put rn=m=n am.
    1. Prove that amrm++ anrn>1- rnrm if m<n, and deduce that anrn diverges.
    2. Prove that anrn<2 (rn-rn+1) and deduce that anrn converges.
  13. Prove that the Cauchy product of two absolutely convergent series converges absolutely.
  14. If {sn} is a complex sequence, define its arithmetic means σn by σn= s0+s1++sn n+1 (n=0,1,2,).
    1. If limsn=s, prove that limσn=s.
    2. Construct a sequence {sn} which does not converge, although limσn=0.
    3. Can it happen that sn>0 for all n and that lim supsn=, although limσn=0?
    4. Put an=sn-sn-1, for n1. Show that sn-σn= 1n+1k=1n kak. Assume that lim(nan)=0 and that {σn} converges. Prove that {sn} converges. [This gives a converse of (a), but under the additional assumption that nan0.]
    5. Define the last conclusion from a weaker hypothesis: Assume M<, |nan|M for all n, and limσn=σ. Prove that limsn=σ, by completing the following outline: If m<n, then sn-σn= m+1n-m (σn-σm)+ 1n-m i=m+1n (sn-si). For these i, |sn-si| (n-i)M i+1 (n-m-1)M m+2 . Fix ε>0 and associate with each n the integer m that satisfies mn-ε1+ε <m+1. Then (m+1)/(n-m)1/ε and |sn-si|<Mε. Hence lim supn |sn-σ| Mε. Since ε was arbitrary, limsn=σ.
  15. Definition 3.21 can be extended to the case in which the an lie in some fixed Rk. Absolute convergence is defined as convergence of |an|. Show that Theorems 3.22, 3.23, 3.25(a), 3.33, 3.34, 3.42, 3.45, 3.47, and 3.55 are true in this more general setting. (Only slight modifications are required in any of the proofs.)
  16. Fix a positive number α. Choose x1>α, and define x2,x3,x4,, by the recursion formula xn+1=12 (xn+αxn).
    1. Prove that {xn} decreases monotonically and that limxn=α.
    2. Put εn=xn-α, and show that εn+1= εn22xn< εn22α so that, setting β=2α, εn+1<β (ε1β)2n (n=1,2,3,).
    3. This is a good algorithm for computing square roots, since the recursion formula is simple and the convergence is extremely rapid. For example, if α=3 and x1=2, show that ε1/β<110 and that therefore ε5<4·10-16 ,ε6<4· 10-32.
  17. Fix α>1. Take x1>α, and define xn+1= α+xn 1+xn =xn+ α-xn2 1+xn .
    1. Prove that x1>x3>x5>.
    2. Prove that x2<x4<x6<.
    3. Prove that limxn=α.
    4. Compare the rapidity of convergence of this process with the one described in Exercise 16.
  18. Replace the recursion formula of Exercise 16 by xn+1= p-1pxn+ αpxn-p+1 where p is a fixed positive integer, and describe the behavior of the resulting sequences {xn}.
  19. Associate to each sequence a={αn}, in which αn is 0 or 2, the real number x(a)=n=1 αn3n. Prove that the set of all x(a) is precisely the Cantor set described in Sec. 2.44.
  20. Suppose {pn} is a Cauchy sequence in a metric space X, and some subsequence {pni} converges to a point pX. Prove that the full sequence {pn} converges to p.
  21. Prove the following analogue of Theorem 3.10(b): If {En} is a sequence of closed nonempty and bounded sets in a complete metric space X, if EnEn+1, and if limn diamEn=0, then 1En consists of exactly one point.
  22. Suppose X is a nonempty complete metric space, and {Gn} is a sequence of dense open subsets of X. Prove Baire's theorem, namely, that 1Gn is not empty. (In fact, it is dense in X.) Hint: Find a shrinking sequence of neighborhoods En such that EnGn, and apply Exercise 21.
  23. Suppose {pn} and {qn} are Cauchy sequences in a metric space X. Show that the sequence {d(pn,qn)} converges. Hint: For any m,n, d(pn,qn) d(pn,pm)+ d(pm,qm)+ d(qm,qn); it follows that d(pn,qn)- d(pm,qm) is small if m and n are large.
  24. Let X be a metric space.
    1. Call two Cauchy sequences {pn},{qn} in X equivalent if limn d(pn,qn)=0. Prove that this is an equivalence relation.
    2. Let X* be the set of all equivalence classes so obtained. If PX*, QX*, {pn}P, {qn}Q, define Δ(P,Q)= limnd (pn,qn); by Exercise 23, this limit exists. Show that the number Δ(P,Q) is unchanged if {pn} and {qn} are replaced by equivalent sequences, and hence that Δ is a distance function in X*.
    3. Prove that the resulting metric space X* is complete.
    4. For each pX, there is a Cauchy sequence all of whose terms are p; let Pp be the element of X* which contains this sequence. Prove that Δ(Pp,Pq) =d(p,q) for all p,qX. In other words, the mapping φ defined by φ(p)=Pp is an isometry (i.e., a distance-preserving mapping) of X into X*.
    5. Prove that φ(X) is dense in X*, and that φ(X)=X* if X is complete. By (d), we may identify X and φ(X) and thus regard X as embedded in the complete metric space X*. We call X* the completion of X.
  25. Let X be the metric space whose points are the rational numbers, with the metric d(x,y)=|x-y|. What is the completion of this space? (Compare Exercise 24.)

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