Problems on Inequalities and Analysis

[archangelic]

Homework Statement

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)

Homework Equations

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 don't have the slightest idea about it.

 

[HallsofIvy]

Homework Statement

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

Think of "if |x|< r then -r< x< r" as solving an inequality rather than a "proof".
The best way to solve inequalities is to first solve the associated equality.
What are the values of x that satisfy |x|= r? There are two roots that divide the line into three intervals. |x|< r will be uniformly true or false in each interval. Check one value of x in each interval to see whether the inequality is true or false in that interval.

Now, the other way. Break -r< x< r into two cases.
a) Suppose 0\le x&lt; r. Then x is non-negative so |x|= x. That makes it pretty easy to show that |x|&lt; r, doesn&#039;t it?<br /> <br /> b) Suppose -r&lt; x&lt; 0. Then x is negative so |x|= -1. What do you get by multiplying each part of -r&lt; x&lt; 0 by -1?<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 2) Solve &amp; write the result in interval notation:<br /> |3 / (2x-1)| &lt; 1 </div> </div> </blockquote><br /> What are the roots of |3/(2x-1)|= 1? You should consider the cases 2x-1&gt; 0 and 2x-1&lt; 0. After you have found the roots of that equation, do the same as in (1).<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> 3) Let S (not a null set) be a bonded(below &amp; above) subset of R. Denote a = inf S, b = sup S. Is it true that &quot;for every &#039;n&#039; (element of N) there exists an &#039;x&#039; (element of S) such that:<br /> a) a &gt; x - (1/n)<br /> b) b &lt; x + (1/n) </div> </div> </blockquote> (a) a&gt; x- (1/n) is the same as a+(1/n)&gt; x. Suppose there were NO x satisfying that. What would that tell you about a+(1/n)? Remember that a is the <b>greatest</b> lower bound.<br /> <br /> (b) Pretty much the same thing. b&lt; x+ (1/n) is the same as x&gt; b- (1/n). If there were NO x satisying that, what would be true of b- (1/n)?<br /> <br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> <h2>Homework Equations</h2><br /> <br /> I have no idea whether any equations could be used to solve this kind of a proof based problem. <br /> <br /> <h2>The Attempt at a Solution</h2><br /> <br /> 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 &quot;&lt;=&gt;&quot; operator.<br /> <br /> Second question, I don&#039;t know how the problem solving approach changes when doing in the perspective of real analysis.<br /> -1 &lt; 3 / (2x-1) &lt; 1<br /> -2x+1 &lt; 3 &lt; 2x-1<br /> x&gt; -1/4 and in the interval notation: (-1/4, infinity)<br /> Hope that&#039;s the way. Please enlighten me if it is to be attempted in some other way.<br /> <br /> Third question, I don&#039;t have the slightest idea about it. </div> </div> </blockquote>

 

[archangelic]

Thanks a lot for that...

Forgive me for asking, but I didn't quite understand this part:

"Now, the other way. Break -r< x< r into two cases.
a) Suppose 0\le x&lt; r. Then x is non-negative so |x|= x. That makes it pretty easy to show that |x|&lt; r, doesn&#039;t it?&quot;<br />

<br /> Thank you once again...