Let $R$ be a ring and suppose that for every $n\in\Bbb Z$ we have a split exact sequence of $R$-modules: $$\{0\}\to E_{n+1}\xrightarrow{\varepsilon_n}E_n\xrightarrow{\pi_n}Q_n\to\{0\}$$ I claim that we have a split exact sequence: $$\{0\}\to\varprojlim_{n\in\Bbb Z}E_n\to\varinjlim_{n\in\Bbb Z}E_n\to\bigoplus_{n\in\Bbb Z}Q_n\to\{0\}$$ First I note that, by induction on $m-n$, that for every $m\geqslant n$ we have the split exact sequence: $$\{0\}\to E_m\to E_n\to\bigoplus_{n\leqslant i\lt m}Q_i\to\{0\}\tag 1$$ where $\varepsilon_{n,m}=\varepsilon_{m-1}\cdots\varepsilon_n:E_m\to E_n$ while the homomorphism $E_n\to\bigoplus_iQ_i$ has components $\varrho_{n,i}\pi_i$, where $\varrho_{n,i}=\varrho_n\cdots\varrho_i$ and $\varepsilon_n\varrho_n=1$ for every $n\in\Bbb Z$. Let $K=\varprojlim_nE_n$ with limit cone $\varkappa_n:K\to E_n$, $L=\varinjlim_nE_n$ with colimit cocone $\lambda_n:E_n\to L$, $N=\bigoplus_iQ_i$. Then I defined $$\{0\}\to K\xrightarrow\varphi L\xrightarrow\psi N\to\{0\}$$ where $\varphi=\varkappa_i\lambda_i$ for every $i\in\Bbb Z$ while $\psi$ is the only morphism such that $\lambda_n\psi$ has components $\varrho_{n,i}\pi_i$ for every $i\geqslant n$. I proved that $\varphi\psi=0$, but how can I prove that $\operatorname{im}(\varphi)\supseteq\ker(\psi)$?
Alternatively, I have also considered to apply functors $\varprojlim_m$ and $\varinjlim_n$ to sequence (1) by taking in account that they preserves split exact sequences... What about this approach?