You can simply estimate $ \left| \sum_{n=1}^\infty \phantom{|} f_n g_n\right| \leq \sum_{n=1}^\infty \phantom{|}|f_n|\,|g_n| \leq K \cdot \sum_{n=1}^\infty\phantom{|} |g_n|. $ This gives you that $h(x) = \sum_{n=1}^\infty f_n(x)g_n(x)$ is well-defined for all $x$ and that the sum converges absolutely and uniformly.
More detail for uniform convergence: By hypothesis there is for each $\varepsilon \gt 0$ an $N$ such that for all $x$ we have $\sum_{n = N+1}^\infty |g_n(x)| \lt \varepsilon/K$. Now for all $x$ and all $M \geq N$ this gives $ \left|h(x) - \sum_{n = 1}^{M} \phantom{|} f_n(x) g_n(x)\right| \leq \sum_{n=N+1}^\infty \phantom{|}|f_n(x)|\,|g_n(x)| \leq K \cdot \sum_{n = N+1}^\infty \phantom{|}|g_n(x)| \lt \varepsilon, $ which shows uniform convergence of the series.