By S. Ponnusamy (auth.)
Mathematical research is prime to the undergraduate curriculum not just since it is the stepping stone for the examine of complex research, but in addition due to its purposes to different branches of arithmetic, physics, and engineering at either the undergraduate and graduate levels.
This self-contained textbook contains 11 chapters, that are extra divided into sections and subsections. each one part features a cautious choice of targeted subject matters lined that would serve to demonstrate the scope and tool of varied equipment in actual research. The exposition is built with thorough factors, motivating examples, routines, and illustrations conveying geometric instinct in a delightful and casual sort to assist readers snatch tough concepts.
Foundations of Mathematical Analysis is meant for undergraduate scholars and starting graduate scholars drawn to a primary creation to the topic. it can be utilized in the school room or as a self-study advisor with none required prerequisites.
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Additional resources for Foundations of Mathematical Analysis
Example text
For instance, using the squeeze rule, we easily have the following: (a) limn→∞ cos n2 /n = 0, because −(1/n) ≤ cos n2 /n ≤ 1/n. With the same reasoning, one has sin(nπ/2) lim = 0. n→∞ n √ √ √ √ √ n + 1 − n = 0 and limn→∞ n( n + 1 − n) = 1/2. (b) limn→∞ Moreover, √ √ 1 1 √ < √ . 2 n n+1+ n √ Note that n. 25. 875. 4 4 4 4 0< n+1− n= √ (c) limn→∞ n/2n = 0. Indeed, using induction we easily see that 2n ≥ n2 for n ≥ 4, so that n 1 0< n ≤ . 2 n √ √ √ 2 (d) limn→∞ bn = 1 if bn = 1/( n + 1) + 1/( n2 + 2) + · · · + 1/( n2 + n).
C) It follows that n! 1 0< n ≤ . n n The second inequality is true because n! = n(n − 1) · · · 2 · 1 < n · n · · · n · 1 = nn−1 . The squeeze rule (with an = 0, cn = 1/n) gives the desired conclusion. 19. 18 may be obtained as a special case of case (b). For instance, if a ≥ 1, then for n large enough we have 1 ≤ a < n. Taking roots on both sides, we obtain 1 ≤ a1/n < n1/n for large n. Again, by the squeeze rule, we see that limn→∞ a1/n = 1. 18 and the product rule for sequences, we can easily obtain that √ lim (3 n)1/2n = 1.
12. If f : A → B and g : B → C are such that f (A) = B and g(B) = C, should (g ◦ f )(A) = C? 13. Which one of the following is not true? (a) f (A ∪ B) = f (A) ∪ f (B). (b) f −1 (A ∪ B) = f −1 (A) ∪ f −1(B). (c) f (A ∩ B) = f (A) ∩ f (B). (d) f −1 (A ∩ B) = f −1 (A) ∩ f −1(B). 14. Can a finite set be equivalent to a proper subset of itself? 15. Must a set be infinite if it is equivalent to a proper subset of itself? 34. 1. Let A consist of all positive rational numbers x whose square is less than 2, and let B consist of all positive rational numbers y such that y 2 > 2.