In the following questions:
Prove that for all real numbers $a$ and $b$, $|a|\leq b$ iff $-b\leq a\leq b$.
Prove that $B\cup(\bigcap \mathcal F)=\bigcap_{A\in \mathcal F}(B\cup A)$.
the answers provided by J. W. Tanner and Brian M. Scott respectively were extremely helpful and after thinking a lot about their answers I gained a new perspective into the mechanics of proof writing $($which currently I am self learning$)$.
Is it a good idea to add those perspectives and explain the root of my mistakes and confusions in the body of the question in form of an edit$?$ Such that it might help future readers$?$