MTH 433 UM Advanced Calculus Analysis with An Introduction to Proof Problems


The text book is Analysis with an introduction to proof 5 edition by steven R There are 6 questions about the text book

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1. [10 points] Mark each statement true or false (you do not need to justify your answer). (a) If
(|sn|) is convergent, then (sn) is convergent. (b) If (sn) is convergent, then (sn) is bounded. (c)
Let S be a subset of R, if S is unbounded, then S has at least one accumulation point. (d) The
union of a finite collection of compact sets is compact. (e) Let S be a subset of R, then S is
closed iff S ′ ⊆ S.
2. [8 points] Consider the set S = { 1 n : n ∈ N}, answer the following questions. (a) Find intS
and justify your answer. (b) Find bdS and justify your answer. (c) Determine if S is open, closed,
neither, or both, and justify your answer.
3. [8 points] Use the definition of convergent sequences (Definition 4.1.2 in textbook; Definition
1.2 in Section 4.1 from class notes) to prove lim n→+∞ n + 3 n2 − 16 = 0.
4. [8 points] Use the Definition 4.2.9 in textbook (Definition 2.9 in Section 4.2 from class notes)
to prove that lim n→+∞ n 2 − n + 1 n + 1 = +∞.
5. [8 points] Prove that the sequence defined below is monotone and bounded. Then find the
limit. s1 = 1 and sn+1 = √ 2sn + 2 for n ≥ 1.
6. [8 points] Suppose that (sn) converges to s. Prove that (s 2 n ) converges to s 2 by using the
definition of convergent sequences (Definition 4.1.2 in textbook; Definition 1.2 in Section 4.1
from class notes). DO NOT use the product formula of Theorem 4.2.1(c) in textbook.

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Advanced calculus

product formula

convergent sequences

monotone and bounded

formula of Theorem

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