Suppose I have three sequences $(a_n)_{n\ge1}$, $(b_n)_{n\ge1}$ and $(c_n)_{n\ge1}$ on the real line which satisfy:
$a_n = b_n + c_n$
The sequence $a_1,a_2,\ldots$ converges (in R), and all elements of the sequence $(c_n)_{n\ge1}$ are in a bounded subset of the real line (say [0,1]). Then perhaps it holds in general that:
$\lim a_n = \liminf b_n + \limsup c_n$
I haven't been able to track down this or similar results in my textbooks. I have a bit of a messy proof/intuition why this should hold, but I believe it cannot be that complicated. Is there a simple proof of the identity? Is there a counter-example?
Let $B_n=b_n-a_n$, $C_n=c_n$. Then $B_n=-C_n$, so $$\liminf B_n=-\limsup C_n.$$On the other hand, since $a_n$ is convergent it's clear that $$\liminf B_n=\liminf b_n-\lim a_n.$$