I know that $\liminf_{n}(a_n+b_n) \geq \liminf_{n}(a_n) + \liminf_{n}(b_n)$ for general RV-ed sequences, but I'm trying to figure out whether we get equality when $(b_n)$ converges, but I'd like to make sure my proof works.
If $\liminf_n (a_n) = -\infty$, this is obvious, so suppose that $\liminf_n(a_n) < \infty$ . Let $\epsilon > 0$. For each $n$, there exists a $j_n \geq n$ such that $a_{j_n} < \inf_{k \geq n} a_k + \epsilon$. Then: $$a_{j_n}+b_n < \inf_{k \geq n} a_k + b_n + \epsilon$$ $$\implies \inf_{k \geq n} (a_{j_k}+b_k) < \inf_{k \geq n} a_k + b_n + \epsilon.$$ Since this holds for each $n$, taking a limit yields: $$\lim_{n \rightarrow \infty}(\inf_{k \geq n} (a_{j_k}+b_k) \leq \lim_{n \rightarrow \infty} (\inf_{k \geq n} a_k + b_n + \epsilon)$$ $$\implies \liminf_{n}(a_n+b_n) \leq \liminf_{n}(a_n)+\lim_n(b_n) + \epsilon.$$ Since $\epsilon > 0$ was arbitrary, we have the desired result.
Does this proof track? It seems to hold, but I'd like to make sure. Thank you.