I am working on a problem from p. 277 of Buck's Advanced Calculus. It asks us to prove the following theorem, without making the change of variable $x=1/t$:
"Let $ \sum_{1}^\infty u_n(x)$ converge to $F(x)$, uniformly for all $x$ with $c\leq x < \infty$.Let $\lim_{x\to\infty}u_n(x) = b_n<\infty$ for $n=1,2,\dotsc$. Then, $\sum_{1}^\infty b_n$ converges, and $\lim_{x\to \infty}F(x) = \sum_1^\infty b_n$."
(The proof with $x=1/t$ consists of employing the theorem that if $f_n$ are continuous on $\overline{S}$ and converge uniformly on the interior of $S$, then they converge uniformly on $\overline{S}$.)
My idea so far has been to take $N$ such that $\sum_{k=N+1}^\infty u_k(x)<\epsilon$ for all $x\geq c$. Now I'd like to find $\kappa \in \mathbb{R}$ such that $|u_k(x)-b_k|<\epsilon_2 \,\,\,\,\forall k\in \mathbb{Z}^+\,\,\forall x\geq \kappa$. But such a point is not available to me, as far as I know. Any ideas?
For the first question, it is enough to show that the sequence $\left(\sum_{n=1}^Nb_n,N\geqslant 1\right)$ is Cauchy. Fix $\varepsilon>0$, and $N_0$ such that if $M\geqslant N\geqslant N_0$, we have $\sup_{x\geqslant c}\left|\sum_{n=N}^Mu_n(x)\right|<\varepsilon$ (by the assumption of uniform convergence). This gives that for such $M,N$, we have $\left|\sum_{n=N}^Mb_n\right|\leqslant \varepsilon$.
For the second question, fix $\varepsilon>0$. Then take an integer $N$ such that for any $x\geqslant c$, $$\left|F(x)-\sum_{n=1}^Nu_n(x)\right|<\varepsilon\quad\mbox{and}\quad\left|\sum_{n\geqslant N+1}b_n\right|<\varepsilon.$$ Then for such $x$, $$\left|F(x)-\sum_{n=1}^{+\infty}b_n\right|+\sum_{n=1}^N|u_n(x)-b_n|\leqslant 2\varepsilon,$$ hence $$\limsup_{x\to +\infty}\left|F(x)-\sum_{n=1}^{+\infty}b_n\right|\leqslant 2\varepsilon.$$ As $\varepsilon$ was arbitrary, we can conclude.