Section 12 12 2 Prove limsup sn = 0 if and only if lim sn = 0 12 4 Show limsup( sn + tn) ≤ lim sup sn + lim sup tn for bounded sequences (sn) and (tn) Hint
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[PDF] Math 3150 Fall 2015 HW2 Solutions
(b) Show that if L > 1, then limsn = +∞ Solution (a) Define the sequence rn = \ \ \ sn+1
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(c) Prove that if limsn and limtn exist, then limsn ≤ limtn Solution (a) Let M > 0 Since sn → +∞ (b) Show that if L > 1, then limsn = +∞ Hint: Apply (a) to the
[PDF] HOMEWORK 5 - UCLA Math
sn ) ≤ lim m→∞ ( inf n>m tn ) = lim inf tn A similar proof works for lim sup Problem 7 (12 2) Prove that lim supsn = 0 if and only if lim sn = 0 Solution If lim
[PDF] 4 Sequences 41 Convergent sequences • A sequence (s n
If (sn) converges to s then we say that s is the limit of (sn) and write s = limn sn, or Proof (a) (⇐) If (sn) is a sequence in S with limit x, and if ϵ > 0 is given, then
[PDF] Problem Sheet 6 1) a) Show that if lim sn = ∞, then lim 1 sn = 0 b
a) a problem from the previous sheet b) the Squeeze Theorem 4) Prove that if sn ≤ tn for all n and lim n→∞ sn = ∞, then lim n→∞ tn = ∞ 5) Use the Squeeze
[PDF] Math 104: Introduction to Analysis SOLUTIONS Alexander Givental
Let a, b ∈ R Show that if a ≤ b+ 1 Assume all sn = 0 and that the limit L = lim sn+1/sn exists Prove that (sn) is bounded if and only if lim sup sn < +∞
[PDF] Homework 3
lim bn = s Prove lim sn = s This is called the “squeeze lemma ” (b) Suppose (sn) and (tn) are (a) Show that if sn ≥ a for all but finitely many n, then lim sn ≥ a
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Section 12 12 2 Prove limsup sn = 0 if and only if lim sn = 0 12 4 Show limsup( sn + tn) ≤ lim sup sn + lim sup tn for bounded sequences (sn) and (tn) Hint
[PDF] Solutions for Homework Math 451(Section 3, Fall 2014)
8 4) Claim: If (tn) is a bounded sequence and (sn) is a sequence such that limsn = 0, then (Alternatively, we could simply prove directly that lim−tn = 0 ) Then
[PDF] Limits
These examples should suggest a result: Theorem 2 20 Let (sn) be a sequence Then lim inf sn = lim sup sn if and only if lim sn = s for
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Homework 6
1.Section 11
11.8 Use Denition 10.6 and Exercise 5.4 to pro velim infsn=limsup(sn) for every sequence (sn). 11.9 (a) Sho wthe closed in terval[ a;b] is a closed set. (b) Is there a sequence (sn) such that (0;1) is its set of subsequential limits?2.Section 12
12.2Pro velim supjsnj= 0 if and only if limsn= 0.
12.4 Sho wlim sup(sn+tn)limsupsn+ limsuptnfor bounded sequences (sn) and (tn).Hint: First show supfsn+tnn > Ng supfsn:n > Ng+ supftn:n > Ng. Then apply Exercise 9.9(c). 12.6 Let ( sn) be a bounded sequence, and letkbe a nonnegative real number. (a) Prove limsup(ksn) =klimsupsn. (b) Do the same for liminf.Hint:Use Exercise 11.8. (c) What happens in (a) and (b) ifk <0? 12.8Let ( sn) and (tn) be bounded sequences of nonnegative numbers. Prove limsupsntn(limsupsn)(limsuptn).
12.9 (a) Pro vethat if lim sn= +1and liminftn>0, then limsntn= +1. (b) Prove that if limsupsn= +1and liminftn>0, then limsupsntn= +1. (c) Observe that Exercise 12.7 is the special case of (b) wheretn=kfor alln2N. 12.10 Pro ve( sn) is bounded if and only if limsupjsnj<+1. 12.11