Did I miss anything in my proof?

[Darya]

Hi, guys. Self-studying real analysis here to prepare for university. I need some feedback if my proof can be qualified as valid cause there's nowhere I could check. Thanks!



Direct proof:
Define set [MATH]A:=(b;+∞)[/MATH]. [MATH]b_1€A[/MATH], because [MATH]b_1>b[/MATH].
Inf(A)=b, as A is bounded below. (should I state why it is b too?)
[MATH]a<=b_1[/MATH], so a is a lower bound. By definition, any lower bound <= greatest lower bound, so a<=b.

(+ question, in the book it wasn't stated what infimum is yet, so can I prove the statement without this concept?)

 

[Steven G]

Inf(A) = b as A is bounded below. Just because A is bounded below does not mean that b is the greatest lower bound---you need to show why b = inf(A) !!

a<=b1 so a is a lower bound. Not necessarily true! a can equal (b+b1)/2. Then a <= b1 but it is not less than b!