By Hugo D. Junghenn

ISBN-10: 148221928X

ISBN-13: 9781482219289

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**Additional resources for A course in real analysis**

**Sample text**

By the preceding observation, one of the resulting subintervals, call it I1 , contains infinitely many terms of the sequence. Choose one such term, say an1 . Now bisect I1 . Again, one of the resulting subintervals, call it I2 , contains infinitely many terms of the sequence. Choose one such term an2 with n2 > n1 . By repeating this procedure, we produce a subsequence {ank }∞ k=1 of {an } and a sequence of intervals Ik = [ck , dk ], k = 0, 1, . , such that c0 ≤ ck−1 ≤ ck ≤ ank ≤ dk ≤ dk−1 ≤ d0 , and dk+1 − ck+1 = 21 (dk − ck ).

3. 8. 2n k=n+1 1 for all n ≥ 1. k 22 A Course in Real Analysis 4. Establish the following formulas by mathematical induction: n n (a) k = n(n + 1)/2. (b) k=1 n k=1 n 2 (c) k 3 = [n(n + 1)/2] . (d) k=1 n (e) k 2 = n(n + 1)(2n + 1)/6. (2k − 1)2 = n(4n2 − 1)/3. k=1 n (2k − 1)3 = n2 (2n2 − 1). (f) k=1 n (g) √ k=1 n (4k 3 − 6k 2 + 4k − 1) = n4 . (h) k=1 √ 1 √ = n. k−1+ k √ 2k + k(k − 1) − 1 √ = n n. 4 to derive and verify a closed formula for n 2 k=1 (5k − 4) . 6. Use known formulas to calculate (a) 1 · 2 + 2 · 3 + 3 · 4 + · · · + 999 · 1000.

10. Prove Bernoulli’s inequality: (1 + x)n ≥ 1 + nx, n ∈ Z+ , x ≥ −1. The Real Number System 23 11. Use the principle of mathematical induction to prove the following variant: Let n0 ∈ Z and let P (n) be a statement depending on integers n ≥ n0 such that (a) P (n0 ) is true, (b) P (n + 1) is true whenever P (j) is true for all n0 ≤ j ≤ n. Then P (n) is true for every n ≥ n0 . 12. (Prime Factorization). Use the variant of induction in Exercise 11 to prove that every integer n ≥ 2 may be written as a product of powers of prime numbers (for example, 72 = 23 · 32 ).

### A course in real analysis by Hugo D. Junghenn

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