I have a real function, $f(n,m)$, which is not necessarily bounded nor necessarily non-negative, but has point-wise convergence of: \begin{equation*} g(m) = \sum\limits_{n=1}^{\infty} f(n,m) \end{equation*} and $g(m)$ is finite for all integers $m$.
I am examining the interchange of limits in \begin{equation*} \sum\limits_{m=1}^{\infty} \sum\limits_{n=1}^{\infty} f(n,m) \sim \sum\limits_{n=1}^{\infty} \sum\limits_{m=1}^{\infty} f(n,m) \end{equation*} and will settle for bounding the summation within a (possibly infinite) range.
If I define: \begin{equation*} g_{N}(m) = \inf_{K \ge N} \sum\limits_{k=1}^{K} f(k,m) \le g(m) \end{equation*} and \begin{equation*} h_{N}(m) = \sup_{K \ge N} \sum\limits_{k=1}^{K} f(k,m) \ge g(m) \end{equation*}
Then, can I state the following?
IF $\sum\limits_{m=1}^{\infty} g(m)$ converges, then the limit is within the (possibly infinite) range of: \begin{equation*} \begin{aligned} \liminf_{N \to \infty} \liminf_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} f(n,m) &\le \lim_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \limsup_{N \to \infty} \limsup_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{N} f(n,m) \end{aligned} \end{equation*}
My reasoning is thus: \begin{equation*} \begin{aligned} \liminf_{N \to \infty} \liminf_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} f(n,m) &\le \liminf_{M \to \infty} \liminf_{N \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{n=1}^{N} f(n,m) \\ &\le \liminf_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{\infty} g_{N}(m) \\ &\le \liminf_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \lim_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \limsup_{M \to \infty} \sum\limits_{m=1}^{M} g(m) \\ &\le \limsup_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{\infty} h_{N}(m) \\ &\le \limsup_{M \to \infty} \limsup_{N \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{N} f(n,m) \\ &\le \limsup_{N \to \infty} \limsup_{M \to \infty} \sum\limits_{m=1}^{M} \sum\limits_{N=1}^{N} f(n,m) \end{aligned} \end{equation*}
With the understanding that the $\liminf$ or $\limsup$ operations could diverge to $\pm \infty$ or converge to different values, the statement above is similar to Fatou's Lemmas, but with the counting measure and with a much, much weaker convergence statement.
My question:
Is this weak, weak convergence statement true for all such $f(n,m)$? If it is not true, can a counter example be provided?