Assuming that $a_{n}$ and $b_{n}$ are two bounded sequences of real numbers, I need to prove that $\inf(a_{n} + b_{n}) \leq \inf(a_{n}) + \sup(b_{n})$.
I have seen proofs that that $\inf(a_{n}+b_{n}) \leq \inf(a_{n}) + \inf(b_{n})$ through this post. Am I missing something obvious here? I'm having a hard time finding an example for the inequality to hold true.
On
Assume on the contrary that $\inf(a_{n}+b_{n})>c>\inf(a_{n})+\sup(b_{n})$. $(1)$
Now take a subsequence of $a_{n}$, say $a_{n_{k}}\to\,\inf(a_{n})$.
By $(1)\,\,\, a_{n_{k}}+b_{n_{k}}>c>\inf(a_{n})+\sup(b_{n})$. Taking $\liminf$
we get $\inf(a_{n})+\liminf\,b_{n_{k}}\,\geq\,c\,>\inf(a_{n})+\sup(b_{n})$
which implies $\liminf\,b_{n_{k}}\,>\,\sup(b_{n})$ which implies that there is a further subsequence $b_{n_{k_{m}}}$
converging to a number greater than $\sup(b_{n})$ , contradiction.
As pointed out by GEdgar, the inequality $\inf(a_{n}+b_{n}) \leq \inf(a_{n}) + \inf(b_{n})$ may fail, for example if $a_n=(-1)^n$ and $b_n=-a_n$.
In order to get the inequality you want, let $B=\sup(b_k)$. Then for each $n$, $a_n+b_n\leqslant a_n+B$. Take the infimum and find a link between $\inf(a_n+B)$, $B$ and $\inf(a_n)$.