Math 441/541 Real Analysis Homework # 1
1. 数学实分析作业代写
Prove that every non-empty subset of R that is bounded above has a least upper bound if and only if every non-empty subset of R that is bounded below has a greatest lower bound. (That is, prove that the Least Upper Bound Property of R is equivalent to the Greatest Lower Bound Property of R)
2.
If A and B are subsets of R, define
A + B = {a + b : a ∈ A and b ∈ B}.
(a) If A and B are both non-empty subsets of R which are bounded above, prove that
sup(A + B) = sup A + sup B.
(b) If A and B are both non-empty subsets of R which are bounded below, prove that
inf(A + B) = inf A + inf B.
3. 数学实分析作业代写
If A and B are both non-empty subsets of R, with A ⊆ B, prove that
inf B ≤ inf A ≤ sup A ≤ sup B.
4.
If A is a non-empty subset of R, define −A = {−a : a ∈ A}. Prove that
sup(−A) = − inf A, and inf(−A) = − sup A.
5. 数学实分析作业代写
Let S be a non-empty subset of R. Let f : S → R and g : S → R be two functions with bounded range; that is, f(S) and g(S) are bounded subsets of R. Prove
(a) sup{f(x) + g(x) : x ∈ S} ≤ sup{f(x) : x ∈ S} + sup{g(x) : x ∈ S}.
(b) inf{f(x) + g(x) : x ∈ S} ≥ inf{f(x) : x ∈ S} + inf{g(x) : x ∈ S}.
(c) Give examples to show that it is possible to have strict inequalities in (a) and (b).
