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EXERCISES - SUP - INF - MAX - MIN

A = {(-1)n/(2+n2) | n ∈ ℕ}; (-1)n/(2+n2) = {1/(2+n2) | n = 2k, k ∈ ℕ; -1/(2+n2) | n = 2k+1, k ∈ ℕ; A = Aeven ∪ Aodd; Aeven = {1/(2+n2) | n = 2k, k ∈ ℕ}; Aodd = {-1/(2+n2) | n = 2k+1, k ∈ ℕ}; sup A = max {sup Aeven, sup Aodd}; inf A = min {inf Aeven, inf Aodd}; 1/(2+n2) ≥ 0 | n = 2k, k ∈ ℕ; -1/(2+n2) ≤ 0 | n = 2k+1, k ∈ ℕ; sup A = sup Aeven; inf A = inf Aodd; 0 ≤ 1/(2+n2) ≤ 1 | n = 2k, k ∈ ℕ; Aeven = {1/2,1/6,1/18...}, 1/(2+n2) decreases when n increases ⇒ sup Aeven = max Aeven = 1/2 ⇒ sup A = sup Aeven = 1/2 = max A; -1 ≤ -1/(2+n2) ≤ 0 | n = 2k+1, k ∈ ℕ; Aodd = {-1/3,-1/11,-1/27...}, -1/(2+n2) increases when n increases; inf Aodd = inf A = -1/3 = min A

A = {|3-n|/(3+n) | n ∈ ℕ}; |x| = {x if x ≥ 0 ∀ x ∈ ℝ; -x if x < 0 ∀ x ∈ ℝ}; |3-n|/(3+n) = {(3-n)/(3+n) if 3-n ≥ 0 ⇔ -n ≥ -3 ⇔ n ≤ 3; (n-3)/(n+3) if 3-n < 0 ⇔ -n < -3 ⇔ n > 3}; 0 ≤ |3-n|/(n+3); |x| ≥ 0 ∀ x ∈ ℝ, |x| > 0 if x ≠ 0; n = 3 ⇒ (3-n)/(n+3) = 0, 0 is a minorant, 0 ∈ A, min A = 0; n = 0 → (3-n)(3+n) = 1 (max), n = 1 → (3-n)(3+n) = 1/2, n = 2 → (3-n)(3+n) = 1/5, n = 3 → (3-n)(3+n) = 0; 0 ≤ (n-3)/(n+3) ≤ 1; n-3 ≤ n+3 ⇒ (n-3)/(n+3) < 1; sup A = max A = 1

A {(-1)n+1/2n | n ∈ ℕ}; (-1)n+1/2n = 1+1/2n | n = 2k, k ∈ ℕ; -1+1/2n | n = 2k+1, k ∈ ℕ; A = Aeven ∪ Aodd; 1+1/2n ≥ 0; -1+1/2n ≤ 0; sup A = sup Aeven; inf A = inf Aodd

; 1+1/2n ≤ 2; Aeven = {2,5/4,17/16...}, sup Aeven = 2 ⇒ sup A = max A = 2; -1 ≤ -1+1/2n ≤ 0; Aodd = {-1/2,-7/8,-31/32...}; inf Aodd = -1 ⇒ inf A = -1; -1 is a minorant; ∀ ε > 0 ∃ n = 2k+1 | -1+ε > -1+1/2n (-1+ε is not a minorant), -1+ε > -1+1/2n ⇔ ε > 1/2n ⇔ 2n > 1/ε true for large values of n; -1 = min A ? ⇔ ∃ n = 2k+1 | -1 = -1+1/2n ⇔ 0 = 1/2n false, ∄ min A

A ⊆ (1,4]; a) inf A = 1; b) A has no minimum; c) A has a majorant; d) A is composed by a finite number of elements; a) false, A = [2,3], A ⊆ (1,4], inf A = 2; b) false, A = [2,3], A ⊆ (1,4], min A = 2; d) false, A = [2,3] is not composed by a finite number of elements; c) true, A ⊆ (1,4] = {x ∈ ℝ | 1 < x ≤ 4} ⇒ 4 is a majorant of A