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131A Group Worksheet 3

January 16, 2024

Problem 1. Find, with proof, sup S, inf S, and max S, min S (if they exist) for the following sets S.

(a) S = [0, 1].

(b) S = [0, 1].

(c) S = {n/1: n ∈ N}.

(d) S = Q ∩ (0, 1). (Hint: first use the Archimedean property to show that there exists a rational number between any two rational numbers.)

Problem 2. Recall the definition of a limit of a sequence: we say that the limit of a sequence  exists and equals L if for every ϵ > 0, there exists an N ∈ N such that

Moreover, we write .

(a) Prove that .

(b) Prove that if  are two sequences with lim, then

(c) Compute