Mathematics 236, Fall 2022 HW #2 Due by 23:59 on October 5 PROBLEM 1. Prove, from the definition of convergence of sequences, that limn→∞2n + 1
You must use an ε – N argument and refrain from using any limit laws.
3n + 2=23.
PROBLEM 2. Suppose (bn) and (xn) are two sequences of real numbers such that (bn) is bounded and (xn) converges to 0. Prove that the sequence (bnxn) converges to 0.
PROBLEM 3. Let (sn)∞n=1 be a convergent sequence of real numbers and suppose α, β ∈ R with α ≤ β. [Limit theorems are ‘off limits’ in parts (a) through (c), but OK to use in part (d).]
(a) Show that if sn ≥ α for all but finitely many n, then lim sn ≥ α.
(b) Show that if sn ≤ β for all but finitely many n, then lim sn ≤ β.
(c) Conclude that if sn ∈ [α, β] for all but finitely many n, then lim sn ∈ [α, β]. (d) Give an example of a convergent sequence (sn) such that sn ∈ (0, 1) but lim sn ∈/ (0, 1).
[ Saying that a statement that depends on n holds true for all but finitely many n means that the set of those n for which the statement is not true is a finite set. ]
PROBLEM 4. Let S be a nonempty bounded subset of R. Prove that there is an increasing sequence (xn)∞n=1 such that xn ∈ S for every n ∈ N and limx→∞xn = sup S.
PROBLEM 5 (an ancient method for the calculation of square roots).
Suppose a is a positive real number, and construct a sequence of real numbers by induction: starting with an arbitrary s1 > 0, define sn+1 := 12(sn + a/sn) for n = 1, 2, 3, . . ..
(a) Prove that s2n ≥ a for every n ≥ 2. Hint: the number sn−1 is a real root of x2 − 2snx + a. (b) Prove that (sn) is eventually decreasing: sn ≥ sn+1 for n ≥ 2.
(c) Prove that (sn) converges to a real number s ≥ 0 such that s2 = a.