Get Basic analysis: Introduction to real analysis PDF

By Jiri Lebl

A primary direction in mathematical research. Covers the genuine quantity process, sequences and sequence, non-stop services, the by-product, the Riemann quintessential, sequences of features, and metric areas. initially constructed to educate Math 444 at college of Illinois at Urbana-Champaign and later greater for Math 521 at collage of Wisconsin-Madison. See http://www.jirka.org/ra/

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6. That is, suppose that {xn } is a bounded sequence and {xnk } is a subsequence. Prove lim inf xn ≤ lim inf xnk . 7. 5: a) Let xn := b) Let xn := (−1)n , find lim sup xn and lim inf xn . n (n − 1)(−1)n , find lim sup xn and lim inf xn . 3. 6: Let {xn } and {yn } be sequences such that xn ≤ yn for all n. Then show that lim sup xn ≤ lim sup yn n→∞ n→∞ and lim inf xn ≤ lim inf yn . 7: Let {xn } and {yn } be bounded sequences. a) Show that {xn + yn } is bounded. b) Show that (lim inf xn ) + (lim inf yn ) ≤ lim inf (xn + yn ).

2 2 As |y − x| < ε for all ε > 0, then |y − x| = 0 and y = x. Hence the limit (if it exists) is unique. 7. A convergent sequence {xn } is bounded. Proof. Suppose that {xn } converges to x. Thus there exists a M ∈ N such that for all n ≥ M we have |xn − x| < 1. Let B1 := |x| + 1 and note that for n ≥ M we have |xn | = |xn − x + x| ≤ |xn − x| + |x| < 1 + |x| = B1 . The set {|x1 | , |x2 | , . . , |xM−1 |} is a finite set and hence let B2 := max{|x1 | , |x2 | , . . , |xM−1 |}. Let B := max{B1 , B2 }.

Now take M := max{M1 , M2 }. For n ≥ M (so that both n ≥ M1 and n ≥ M2 ) we have |y − x| = |xn − x − (xn − y)| ≤ |xn − x| + |xn − y| ε ε < + = ε. 2 2 As |y − x| < ε for all ε > 0, then |y − x| = 0 and y = x. Hence the limit (if it exists) is unique. 7. A convergent sequence {xn } is bounded. Proof. Suppose that {xn } converges to x. Thus there exists a M ∈ N such that for all n ≥ M we have |xn − x| < 1. Let B1 := |x| + 1 and note that for n ≥ M we have |xn | = |xn − x + x| ≤ |xn − x| + |x| < 1 + |x| = B1 .

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Basic analysis: Introduction to real analysis by Jiri Lebl


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