If $\{a_n\}$ is a series, and $\lim(a_{n+1} - a_n) = 0$, prove that $P(a_n) = [\lim\inf(a_n),\lim\sup(a_n)]$
Basically, I wish to prove that the set of all partial limits of the series $a_n$ (as $n\to\infty$) contains all values between its maximal limit ($\lim\sup$) and minimal limit ($\lim\inf$)
I already know that $\lim\sup$ and $\lim\inf$ are limits of $a_n$ by definition, and so it is a closed set $[\lim\inf(a_n),\lim\sup(a_n)]$.
Since $\lim\sup(a_n) = \sup(P(a_n))$ and $\lim\inf(a_n) = \inf(P(a_n))$ then $P(a_n)$ is included in $[\lim\inf(a_n),\lim\sup(a_n)]$.
Now if I were to take any $\lim\inf(a_n) < L < \lim\sup(a_n)$, I need to prove that it is a partial limit.