If you have a sequence $ a_n $ which is bounded and monontonic increasing the theorem tells us it converges to a limit $ L $
But having looked at the proof is it implicit that this limit $ L= Sup \left \{a_n : n \in \mathbb{N} \right \} $ ?
and does the proof also implicitly tell you if $ a_n $ is bounded and monontonic decreasing then $ L= Inf \left \{a_n : n \in \mathbb{N} \right\} $ ?
I.e asking to show that $ L= Sup/Inf \left \{a_n : n \in \mathbb{N} \right \} $ is the same as asking for proof of bounded and monontonic sequence theorem? (edit: given the sequence is bounded and monotone)