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Suppose $A$ is a nonempty bounded above set, containing two variables, say $n$ and $m$. If I want to find $\sup A$ and $\inf A$, does the following procedure work?

$(1)$ Find an upper bound for $A$.

$(2)$ Find an increasing subsequence that converges to the upper bound

Then, the upper bound is the supremum for $A$.

Same goes to finding $\inf A$, by replacing upper bound with lower bound and increasing with decreasing.

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    You might want nondecreasing instead of increasing to account for the case where the supremum is already in the set. – 2017-01-12

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The procedure indeed yields the supremum if you manage to carry out both steps - not always an easy thing to do!

The reason is that one definition of supremum is "an upper bound smaller than every other upper bound". If you find an upper bound $x$, and find a subsequence converging to it, it means that anything smaller than $x$ is eventually surpassed by the subsequence ... so it can't be an upper bound, i.e. there cannot be an upper bound smaller than $x$.

Same thing for the infimum!

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If $x$ is an upper bound for $A$ and $(a_n)_n$ is a sequence in $A$ converging to $x$ then no $y