辅导案例-MAST20026-Assignment 3

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MAST20026 Real Analysis Semester 2, 2020
Assignment 3
Due: Wednesday, 16th September (11:59pm)
Write your name and your student number in the spaces provided below.
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correct answer space for the question, warning the marker that you have appended additional
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Name:
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The assignment begins on the next page.
Mathematics and Statistics 1 of 3 University of Melbourne
MAST20026 Real Analysis Semester 2, 2020
1. Use Mathematical Induction to prove that for all positive integers n
n∑
j=1
(−1)j(j − 1) · j = 1 · 2− 2 · 3 + ... + (−1)n(n− 1) · n =
{
n2
2 n even
1−n2
2 n odd
.
Mathematics and Statistics 2 of 3 University of Melbourne
MAST20026 Real Analysis Semester 2, 2020
2. Define S = {x ∈ R | ∃y > 0, x = y + 1y}. In the following proofs, clearly state the axioms for R or
theorems used.
(a) Prove that S is not bounded above.
(b) Prove that S is bounded below.
(c) Guess the infimum inf S and prove that your guess is correct.
Mathematics and Statistics 3 of 3 University of Melbourne

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