Let $M$ and $N$ be $R-$modules, and $$0\longrightarrow M\overset{\iota}{\longrightarrow} N\overset{\pi}{\longrightarrow} N/M\longrightarrow 0$$ an exact sequence. Let $M$ and $N/M$ finitely generated. Show that $N$ is finitely generated.
I know how to prove it, but I wanted to do as following if it's possible. Since $M$ and $N/M$ are finitely generated, then there is a surjection $$\sigma :R^{\oplus s}\longrightarrow M$$ and a surjection $$\tau : R^{\oplus t}\longrightarrow N/M.$$
What I want to show is that there is a surjection $$R^{\oplus s}\oplus R^{\oplus t}\longrightarrow N.$$ If $$\pi_1:R^{\oplus s}\oplus R^{\oplus t}\longrightarrow R^{\oplus s}$$ and $$\pi_2: R^{\oplus s}\oplus R^{\oplus t}\longrightarrow R^{\oplus t}$$ are the projection, we have that $\sigma \circ \pi_1$ and $\tau\circ \pi_2$ are onto, but ow can I do better ?
Hint:
Snake lemma or, if not at your disposal, diagram hunting in \begin{alignat}{4} 0\longrightarrow R^s&\longrightarrow\, & R^s\oplus R^t &\longrightarrow &R^t~~&\longrightarrow 0 \\ \downarrow~&&\downarrow\quad~&&\downarrow~~\\ 0\longrightarrow M &\longrightarrow & N\quad~ &\longrightarrow~ & N/M&\longrightarrow 0 \end{alignat}