Real Analysis Lecture 6 PDF

Theorem Let lim 𝑥𝑛 = 𝑥, then 𝑛→∞ a) If 𝑥𝑛 ≥ 𝑎 ∀𝑛 ∈ ℕ, then 𝑥 ≥ 𝑎 b) If 𝑥𝑛 ≤ 𝑏 ∀𝑛 ∈ ℕ, then 𝑥 ≤ 𝑏 Sandwich Theorem Let lim 𝑥𝑛 = 𝑙 = lim 𝑧𝑛 and 𝑥𝑛 ≤ 𝑦𝑛 ≤ 𝑧𝑛 𝑛 ∈ ℕ, then lim 𝑦𝑛 = 𝑙 𝑛→∞ 𝑛→∞ 𝑛→∞ Proof Since lim 𝑥𝑛 = 𝑙 = lim 𝑧𝑛 given 𝜀 > 0, the...

Theorem Let lim 𝑥𝑛 = 𝑥, then 𝑛→∞ a) If 𝑥𝑛 ≥ 𝑎 ∀𝑛 ∈ ℕ, then 𝑥 ≥ 𝑎 b) If 𝑥𝑛 ≤ 𝑏 ∀𝑛 ∈ ℕ, then 𝑥 ≤ 𝑏 Sandwich Theorem Let lim 𝑥𝑛 = 𝑙 = lim 𝑧𝑛 and 𝑥𝑛 ≤ 𝑦𝑛 ≤ 𝑧𝑛 𝑛 ∈ ℕ, then lim 𝑦𝑛 = 𝑙 𝑛→∞ 𝑛→∞ 𝑛→∞ Proof Since lim 𝑥𝑛 = 𝑙 = lim 𝑧𝑛 given 𝜀 > 0, there exist positive integers 𝑚1 and 𝑚2 such 𝑛→∞ 𝑛→∞ |𝑥 − 𝑙 | < 𝜀 that 𝑛 ∀𝑛 ≥ 𝑚= 𝑚𝑎𝑥{𝑚1 , 𝑚2 } |𝑧𝑛 − 𝑙 | < 𝜀 ⇒ 𝑙 − 𝜀 < 𝑥𝑛 ≤ 𝑦𝑛 ≤ 𝑧𝑛 < 𝑙 + 𝜀 ∀𝑛 ≥ 𝑚 ⇒ 𝑙 − 𝜀 < 𝑦𝑛 < 𝑙 + 𝜀 ∀𝑛 ≥ 𝑚 Hence lim 𝑦𝑛 = 𝑙 𝑛→∞ Theorem 𝑥1+𝑥2 +⋯+𝑥𝑛 If lim 𝑥𝑛 = 0, then lim ( )=0 𝑛→∞ 𝑛→∞ 𝑛 Note the converse of the above theorem need not be true. Monotone sequences Definition A sequence {𝓍𝑛 } is said to be a) monotone increasing (or non-decreasing) if 𝑥𝑛 ≤ 𝑥𝑛+1 ∀𝑛 ∈ ℕ b) Strictly monotone increasing if 𝑥𝑛 < 𝑥𝑛+1 ∀𝑛 ∈ ℕ c) Monotone decreasing (or non-increasing) if 𝑥𝑛 ≥ 𝑥𝑛+1 ∀𝑛 ∈ ℕ d) Strictly monotone decreasing if 𝑥𝑛 > 𝑥𝑛+1 ∀𝑛 ∈ ℕ A sequence is said to be monotone if {𝓍𝑛 } is either monotone increasing or monotone decreasing. Example The sequence {1,2,3,4,4, 4...} is a monotone increasing but not strict. The sequence {n} is strictly monotone increasing. Theorem 1 A monotone increasing and bounded above sequence converges to its supremum. Proof Let {𝓍𝑛 } be a monotone increasing bounded above sequence. Let R= {𝑥1 , 𝑥2 … } be the range set of {𝓍𝑛 }. Since {𝓍𝑛 } is bounded above, R is bounded above. Then by completeness property, R has a supremum say M. Let 𝜀 > 0 be given, then there exist an 𝑥𝑘 ∈ 𝑅 such that 𝑀 − 𝜀 < 𝑥𝑘 ≤ 𝑥𝑛 ∀𝑛 ≥ 𝑘 ({𝓍𝑛 } is a monotone increasing) ≤M (M is the supremum of R) < 𝑀 + 𝜀 (𝜀 > 0) ⇒ 𝑀 − 𝜀 < 𝑥𝑛 < 𝑀 + 𝜀 ∀𝑛 ≥ 𝑘 ⇒ |𝑥𝑛 − 𝑀| < 𝜀 ∀𝑛 ≥ 𝑘 ⇒ {𝓍𝑛 } is convergent and converges to M. Theorem 2 A monotone decreasing and bounded below sequence converges to its infimum. From the above theorem 1 and 2, we get the following theorem: Monotone Convergent theorem. A monotone sequence is convergent if and only if it is bounded Subsequence Definition Let {𝓍𝑛 } be a sequence and {𝑛𝑘 } be a strictly increasing sequence in ℕ. The sequence {𝓍𝑛𝑘 } is called a subsequence of {𝓍𝑛 }. Theorem A sequence {𝓍𝑛 } converges to 𝓍 if and only if every subsequence of {𝓍𝑛 } converges to 𝓍. Proof Let {𝓍𝑛𝑘 } be an arbitrary subsequence of {𝓍𝑛 }. Suppose first that {𝓍𝑛 } converges to 𝓍. Then for each 𝜀 > 0 there exists a positive integer 𝑚 such that |𝑥𝑛 − 𝑥 | < 𝜀 ∀𝑛 ≥ 𝑚. If 𝑘> 𝑚 then 𝑛𝑘 ≥ 𝑘> 𝑚. Therefore, |𝑥𝑛 − 𝑥 | < 𝜀 ∀𝑘> 𝑚. This shows that {𝓍𝑛𝑘 } converges to 𝓍. Then for each 𝜀 > 0 there exists a positive integer N such that |𝑥𝑛𝑘 − 𝑥 | < 𝜀 ∀𝑘 ≥ 𝑁 ⇒ {𝓍𝑛 } converges to 𝑥 Bolzano Weiertrass theorem Every bounded sequence of real numbers has a convergent subsequence. Proof We call the 𝑚𝑡ℎ term of 𝑥𝑚 of the sequence {𝓍𝑛 } a “peak” if 𝑥𝑚 ≥ 𝑥𝑛 ∀𝑛 ≥ 𝑚 Case 1 If {𝓍𝑛 } has infinitely many peaks, order them by increasing subscripts 𝑚1 < 𝑚2 < ⋯ < 𝑚𝑘 … Since each of 𝑥𝑚1 , 𝑥𝑚2 is a peak, we have 𝑥𝑚1 ≥ 𝑥𝑚2 ≥ 𝑥𝑚3 … We thus get a subsequence {𝑥𝑚𝑘 } of {𝓍𝑛 } which is monotone decreasing. Also {𝑥𝑚𝑘 } is bounded since {𝓍𝑛 } is bounded. By monotone convergence theorem, {𝑥𝑚𝑘 } is convergent. Case 2 Let {𝓍𝑛 } has a finite number of peaks namely 𝑥𝑚1 , 𝑥𝑚2 , 𝑥𝑚𝑘 ,... Put 𝑘1 = 𝑚𝑘 + 1. Then 𝑥𝑘1 is not a peak, there exists a 𝑘2 > 𝑘3 such that 𝑥𝑘3 >𝑥𝑘2 Continuing this way, we get a subsequence {𝑥𝑘1 } of {𝓍𝑛 } which is monotone increasing. Also, since {𝓍𝑛 } is bounded, {𝓍𝑘𝑙 } is bounded. Therefore, by monotone convergence theorem, {𝓍𝑘𝑙 } is convergent. Divergent sequences Definition A sequence {𝓍𝑛 } of real numbers is said to a) Diverge to +∞, written lim 𝑥𝑛 = + ∞ if for every real number p, there exists a 𝑛→∞ positive integer G=G(p) (depending on p) however large it may be such that 𝑥𝑛 > 𝑝 ∀𝑛 ≥ 𝐺(𝑝) b) Diverges to -∞, written as lim 𝑥𝑛 = − ∞ if every real number q there exists a 𝑛→∞ positive integer H=H(q) (depending on 1) however small it may be such that 𝑥𝑛 < 𝑞 ∀𝑛 ≥ 𝐻(𝑞) A sequence is said to be divergent to +∞ or diverges to −∞. Examples 1) The sequence {n} diverges to +∞ 2) The sequence {1 − 𝑛2 } diverges to −∞