If $A\subset B$ and $B$ is bounded above, show that $\operatorname{lub}A \leq \operatorname{lub}B$.
This seems very obvious but I am not able to write a proper solution for this. Does it really require a proof or we can logically conclude this? Kindly help.
Yes it requires a proof, by logically concluding.
Let $x$ be the least upper bound of $B$, then it's an upper bound for $A\subseteq B$, hence $\sup A\le x$.