Question

explain this proof in real analysis

Answer 1

What is Def 1.8?

Answer 2

I don't quite understand their explanation either.

We know \(L\) contains all lower bounds of \(B\): \[y \in L\iff \forall x \in B \quad y \leq x\]This consequentially means elements in \(B\) are upper bounds of \(L\):\[
\forall x \in B\quad \forall y \in L \quad y\leq x
\]We know \(B\) is bounded below: \[\exists \lambda\quad \lambda \in L\land \lambda \in B\]Because \(\lambda \in B\), it is an upper bound of \(L\).
Any element \(m<\lambda\) can't be an upper bound of \(L\), because \(\lambda\) is in \(L\).
Any element \(m>\lambda\) can't be the supremum, because the supremum is the minimum upper bound.
Therefore, \(\lambda = \sup L\).