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Okay, so I'm struggling with understanding where I went wrong. The instructor feels like I don't understand the material and when she presented my explanation to a colleague, he too agreed with her.

I would really appreciate if someone could tell me the first part of where I went wrong in my logic. If not, what do I need to re-word or what information do I need to provide?

This is question 1.19 in Apostol. Feel free to be as ruthless as you want. I really do WANT to understand and therefore, if I need a reality check, then I'm willing to have my feelings hurt to pass the test. I'm a graduate engineering student that accidentally picked the wrong math class to fullfill my requirement. I have to admit, I do appreciate what the class is teaching but I am having a hard time at getting it..

19) Find the sup and inf of each of the following sets of real numbers.

c) S = {x: (x-a)(x-b)(x-c)(x-d) < 0} where a < b < c < d

I proved that

x > a

x < d

Therefore, I stated since :

The set is bounded above by d, then sup(S) = d (no upperbound greater than d).

The set is bounded below by a, then inf(S) = a (no lowerbound point less than a).

This was accepted by my professor (so I'm aiming that you can accept this without me uploading my proof).

She then asked, "What about a < x < b".

I said that it was irrelevant because for all x, no matter what a, b, c, d (so long as they meet the given problem statement), the set is bounded above and below.

I also said that because any value between a < x < b cannot exist, it is not in the set, so it doesn't matter anyway (by definition of the problem statement).

I am told I am wrong by two PhDs, so now, I need to dig deep and try to understand.

Maybe my logic is right, but my notation is wrong? May be my logic is right but my understanding is wrong? My logic is wrong? May be it's some combination of all the above and I'm a lost cause?

Please help!

I've attached the PDF file. Any comments on notation or concepts I'm missing is GREATLY appreciated.

I would really appreciate if someone could tell me the first part of where I went wrong in my logic. If not, what do I need to re-word or what information do I need to provide?

This is question 1.19 in Apostol. Feel free to be as ruthless as you want. I really do WANT to understand and therefore, if I need a reality check, then I'm willing to have my feelings hurt to pass the test. I'm a graduate engineering student that accidentally picked the wrong math class to fullfill my requirement. I have to admit, I do appreciate what the class is teaching but I am having a hard time at getting it..

19) Find the sup and inf of each of the following sets of real numbers.

c) S = {x: (x-a)(x-b)(x-c)(x-d) < 0} where a < b < c < d

I proved that

x > a

x < d

Therefore, I stated since :

The set is bounded above by d, then sup(S) = d (no upperbound greater than d).

The set is bounded below by a, then inf(S) = a (no lowerbound point less than a).

This was accepted by my professor (so I'm aiming that you can accept this without me uploading my proof).

She then asked, "What about a < x < b".

I said that it was irrelevant because for all x, no matter what a, b, c, d (so long as they meet the given problem statement), the set is bounded above and below.

I also said that because any value between a < x < b cannot exist, it is not in the set, so it doesn't matter anyway (by definition of the problem statement).

I am told I am wrong by two PhDs, so now, I need to dig deep and try to understand.

Maybe my logic is right, but my notation is wrong? May be my logic is right but my understanding is wrong? My logic is wrong? May be it's some combination of all the above and I'm a lost cause?

Please help!

I've attached the PDF file. Any comments on notation or concepts I'm missing is GREATLY appreciated.