By Jiri Lebl

A primary path in mathematical research. Covers the genuine quantity approach, sequences and sequence, non-stop services, the by-product, the Riemann essential, sequences of capabilities, and metric areas. initially built to coach Math 444 at collage of Illinois at Urbana-Champaign and later more desirable for Math 521 at college of Wisconsin-Madison. See http://www.jirka.org/ra/

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Let x1 , x2 , . . , xn ∈ R. Then |x1 + x2 + · · · + xn | ≤ |x1 | + |x2 | + · · · + |xn | . Proof. We will proceed by induction. Note that it is true for n = 1 trivially and n = 2 is the standard triangle inequality. Now suppose that the corollary holds for n. Take n + 1 numbers x1 , x2 , . . , xn+1 and compute, first using the standard triangle inequality, and then the induction hypothesis |x1 + x2 + · · · + xn + xn+1 | ≤ |x1 + x2 + · · · + xn | + |xn+1 | ≤ |x1 | + |x2 | + · · · + |xn | + |xn+1 | .

5. Let {xn } and {yn } be convergent sequences. (i) The sequence {zn }, where zn := xn + yn , converges and lim (xn + yn ) = lim zn = lim xn + lim yn . n→∞ n→∞ n→∞ n→∞ (ii) The sequence {zn }, where zn := xn − yn , converges and lim (xn − yn ) = lim zn = lim xn − lim yn . n→∞ n→∞ n→∞ n→∞ (iii) The sequence {zn }, where zn := xn yn , converges and lim (xn yn ) = lim zn = lim xn n→∞ n→∞ n→∞ lim yn . n→∞ 50 CHAPTER 2. SEQUENCES AND SERIES (iv) If lim yn = 0, and yn = 0 for all n, then the sequence {zn }, where zn := xn , converges and yn xn lim xn = lim zn = .

N→∞ (ii) If c > 1, then {cn } is unbounded. Proof. First let us suppose that c > 1. We write c = 1 + r for some r > 0. By induction (or using the binomial theorem if you know it) we see that cn = (1 + r)n ≥ 1 + nr. Now by Archimedean property of the real numbers {1 + nr} is unbounded (for any number B, we can find an n such that nr ≥ B − 1). Therefore cn is unbounded. 1 Now let c < 1. Write c = 1+r , where r > 0. Then cn = 1 1 11 ≤ ≤ . n (1 + r) 1 + nr r n As { n1 } converges to zero, so does { 1r n1 }.