I don’t follow what you posted.
Suppose that then if then .
This means that or .
This shows that .
Suppose that then .
That means or .
What is wrong with that?
Theorem
sup exists inf exists
Proof:
exists
Now we show that
Show that
1. is a lower bound of .
2. if is a lower bound of -sup
1.
is an upper bound of
is a lower bound of (line added for correction 6:30 a.m. Sep 12)
2. Suppose that is a lower bound of
suppose not
on the other hand
<------(Could someone tell me how this come about.)
Hence, is an upper bound of
By 1. & 2, & =
My original post was from lecture notes of some school of mathematics I found on the web--I don't remember where I got it. There is a missing link in the proof which made it hard to understand.
Since you said you could not follow it, you made me feel better.
Your instructions are superb, which I am very glad that you are on board.
I am not a math major, so I don't have access to a real math professor, but thanks to MHF for giving me access to a teacher like you.
Thank you, sir.