Noetherian property for exact sequence

This follows immediately from the Five Lemma applied to $$\begin{array}{c} 0 & \longrightarrow & \alpha^{-1}(L_n) & \longrightarrow & L_n & \longrightarrow & \beta(L_n) & \longrightarrow & 0 \\ & & \downarrow && \downarrow && \downarrow & \\ 0 & \longrightarrow & \alpha^{-1}(L_{n+1}) & \longrightarrow & L_{n+1} & \longrightarrow & \beta(L_{n+1}) & \longrightarrow & 0.\end{array}$$

Let $x\in L_{n+1}$. Then $\beta(x)\in\beta(L_{n+1})$ and let's suppose that $\beta(L_n)=\beta(L_{n+1})$. We have $\beta(x)\in\beta(L_n)$, so there is $y\in L_n$ such that $\beta(x)=\beta(y)$ hence $x-y\in\ker\beta$. From $\ker\beta=\operatorname{im}\alpha$ we get $x-y=\alpha(z)$ for some $z\in M'$. Since $x-y\in L_{n+1}$ we have $z\in\alpha^{-1}(L_{n+1})=\alpha^{-1}(L_{n})$, so $x-y=\alpha(z)\in L_n$ and therefore $x\in L_n$.