Show that the space $L^1_{\text{loc}}(\mathbb{R})$ endowed with a system of seminorms $$ q_n(f) = \int_{-n}^n |f| \quad f \in L^1_{\text{loc}}(\mathbb{R}), n \in \mathbb{N} $$ is a Fréchet space.
We already know, that the space is locally convex, we need to check if is is an [F-space][2].
We define a metric as follows $$ \rho(f,g) = \sum_{n=1}^\infty \frac{1}{2^n} \text{min} \{1, q_n(f-g)\} \quad f,g \in L^1_{\text{loc}}(\mathbb{R}). $$ Then the metric $\rho$ is translation invariant. We need to show that $\rho$ is complete.
Choose a $\rho$-cauchy sequence $(f_m)$ in $L^1_{\text{loc}}(\mathbb{R})$. We also know, that the sequence $\left(f_m \vert_{[-n,n]}\right)$ is $q_n$-cauchy for all $n \in \mathbb{N}$. This means that $$ q_n(f) = \lVert f \lvert_{[-n,n]} \rVert_{L^1(-n,n)}. $$ Since $L^1$ is a Banach space, we can find a function $g_n \in \left( L^1([-n,n]), q_n \right)$ such that for all $n \in \mathbb{N}$ $$ f_m \xrightarrow{q_n} g_n, $$ as $m \to \infty$.
Now we need to find $g \in L^1_{\text{loc}}(\mathbb{R})$, such that for all $n \in \mathbb{N}$ $$ q_n(f_m - g) \xrightarrow{m \to \infty} 0. $$
I tried to define $g$ as follows $$ g(x) = \begin{cases} g_n(x) & x \in [-n,n]\\ 0 & x \not\in [-n,n]\\ \end{cases} $$ But this does not work, since $g$ depends on $n$.
How can I proceed from here? Thanks.