I encountered a few problems while attending a problem solving seminar. Abstract mathematics and real analysis is not my forte and haven't really taken any courses in that regard. Thought maybe someone here could offer some help in better understanding about the following topics. Hope I am posting in the right section too.
1) Let r > 0. Prove that: |x| < r <==> -r < x < r
2) Solve & write the result in interval notation:
|3 / (2x-1)| < 1
3) Let S (not a null set) be a bonded(below & above) subset of R. Denote a = inf S, b = sup S. Is it true that "for every 'n' (element of N) there exists an 'x' (element of S) such that:
a) a > x - (1/n)
b) b < x + (1/n)
I have no idea whether any equations could be used to solve this kind of a proof based problem.
The Attempt at a Solution
For the first question, I am quite not sure, whether it is to prove P implies Q or Q implies P or both. This is the first time, with that "<=>" operator.
Second question, I don't know how the problem solving approach changes when doing in the perspective of real analysis.
-1 < 3 / (2x-1) < 1
-2x+1 < 3 < 2x-1
x> -1/4 and in the interval notation: (-1/4, infinity)
Hope that's the way. Please enlighten me if it is to be attempted in some other way.
Third question, I dont have the slightest idea about it.