Suppose we have 2 real power series for every $k=1,2,3,\ldots$ $$ f_k(x)=\sum_{n\in\mathcal{S}^{(+)}(k)} x^n d_n(k),\quad g_k(x)=\sum_{n\in\mathcal{S}^{(-)}(k)} x^n d_n(k), \quad d_n(k)\geq 0 $$ where $\mathcal{S}^{(+)}(k)\cup\mathcal{S}^{(-)}(k)=\{1,2,3,\ldots\}$ and $\mathcal{S}^{(+)}(k)\cap\mathcal{S}^{(-)}(k)=\emptyset.$
We know that they satisfy the following 2 conditions:
Condition 1: $$ 0\leq f_k(x)<\infty,\quad 0\leq g_k(x)<\infty,\quad \forall\; k=1,2,3,\ldots \text{ and } \forall\; 0\leq x<\infty\\ $$ (i.e. they have radius of convergence infinity for every k)
Condition 2: $$ -\infty<\liminf_{k\rightarrow\infty}\big(f_k(x)-g_k(x)\big)\leq\limsup_{k\rightarrow\infty}\big(f_k(x)-g_k(x)\big)<\infty,\quad \forall\; 0\leq x<\infty $$
THE QUESTION: Do conditions 1) and 2) imply $$ \limsup_{k\rightarrow \infty} f_k(x)<\infty,\quad \limsup_{k\rightarrow \infty} g_k(x)<\infty,\quad \forall \, 0\leq x<\infty $$ I've tried proof by contradiction, i.e. suppose (by contradiction) that $\exists$ $0<\gamma<\infty$ such that $$ \limsup_{k\rightarrow \infty} f_k(\gamma)=\infty,\quad \limsup_{k\rightarrow \infty} g_k(\gamma)=\infty, $$ and Condition 2 is satisfied for $x=\gamma$: $$ -\infty<\liminf_{k\rightarrow\infty}\big(f_k(\gamma)-g_k(\gamma)\big)\leq\limsup_{k\rightarrow\infty}\big(f_k(\gamma)-g_k(\gamma)\big)<\infty, $$ (i.e. the 2 infinities cancel each other out) Now let $x\rightarrow x\gamma$ and examine $$ \limsup_{k\rightarrow\infty}\Big( \sum_{n\in\mathcal{S}^{(+)}(k)} x^n (\gamma ^n d_n(k)) - \sum_{n\in\mathcal{S}^{(-)}(k)} x^n (\gamma ^n d_n(k)) \Big),\quad \forall\; 0<x<\infty $$ What I expect is that the two infinities $\big(\;\sum_{n\in\mathcal{S}^{(+)}(k)} \gamma ^n d_n(k)$ and $\sum_{n\in\mathcal{S}^{(-)}(k)} \gamma ^n d_n(k)\,\big)$ cannot cancel each other out for some $0<x<\infty$, i.e. I expect that $\exists$ $0<x<\infty$ such that the above quantity is infinite (and hence contradicting Condition 2). But I'm not sure how to prove it rigorously. Note that since the intersection of the 2 sets $\mathcal{S}^{(+)}(k),$ $\mathcal{S}^{(+)}(k)$ is the empty set, that $f_k$ and $g_k$ never contain the same power in $x$. Also not that the proofs of convergence would be straightforward if we were not taking the limit since then we would have a power series and could apply the well-know results. If you have any ideas/comments or know how to prove this that would be great! Thanks!