1. (10 points) Consider the set2020n + 2021S =:n = 1,2,3, .n+1Find sup(S) and inf(S) and JUSTFIY!2. (10 points) On R? define a function d: R? x R? → [0, 0) given byd (x1, y1), (x2, y2)) = 2020 · |a1 - x2| + 2021 |yı – y2|-%3DShow that d is a metric on R? and sketch the neighborhood N1((0,0)) of the origin with radius 1.3. (15 points) Consider the sequence {rn} given byT1 = v2020 and rn+1 =2020 +an for n = 1,2, 3, ....Prove that {an} is Cauchy.4. (15 points) Define f : (2020, 2021) → R via2021f(x):x - 2020Show that f is NOT uniformly continuous.5. (15 points) Let f : R R be a differentiable function such that f(r) > 2 and |f'(x)| < 2020 for any r.Show that In(f(x)) is uniformly continuous.6. (15 points) Define2020fn(x) = sin ( a +for n = 1,2, 3,....Find the pointwise limit f of {fn} and prove that fn converges to f uniformly.

Name: 1. (10 points) Consider the set2020n + 2021S =:n = 1,2,3, .n+1Find su
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Description: 1. (10 points) Consider the set2020n + 2021S =:n = 1,2,3, .n+1Find sup(S) and inf(S) and JUSTFIY!2. (10 points) On R? define a function d: R? x R? → [0, 0) given byd (x1, y1), (x2, y2)) = 2020 · |a1 - x2| + 2021 |yı – y2|-%3DShow that d is a metric

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1. (10 points) Consider the set 2020n + 2021 S = :n = 1,2, 3, n+1 Find sup(S) and inf(S) and JUSTFIY!

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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