We freely use the results of Lemma \ref{lemma-describe-constructible-complexes}. Choose $a, b$ as in that lemma. We will prove the lemma by induction on $b - a$. Let $\mathcal{F} = H^b(K)$. Note that $\mathcal{F}$ is a derived complete sheaf of $\Lambda$-modules by Proposition \ref{proposition-enough-weakly-contractibles}. Moreover $\mathcal{F}/I\mathcal{F}$ is a locally constant sheaf of $\Lambda/I$-modules of finite type. Apply Lemma \ref{lemma-derived-complete-limit} to get a surjection $\rho : (\underline{\Lambda}^\wedge)^{\oplus t} \to \mathcal{F}$. \medskip\noindent If $a = b$, then $K = \mathcal{F}[-b]$. In this case we see that $$ \mathcal{F} \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I} = \mathcal{F}/I\mathcal{F} $$ As $X$ is weakly contractible and $\mathcal{F}/I\mathcal{F}$ locally constant, we can find a finite disjoint union decomposition $X = \coprod U_i$ by affine opens $U_i$ and $\Lambda/I$-modules $\overline{M}_i$ such that $\mathcal{F}/I\mathcal{F}$ restricts to $\underline{\overline{M}_i}$ on $U_i$. After refining the covering we may assume the map $$ \rho|_{U_i} \bmod I : \underline{\Lambda/I}^{\oplus t} \longrightarrow \underline{\overline{M}_i} $$ is equal to $\underline{\alpha_i}$ for some surjective module map $\alpha_i : \Lambda/I^{\oplus t} \to \overline{M}_i$, see Modules on Sites, Lemma \ref{sites-modules-lemma-morphism-locally-constant}. Note that each $\overline{M}_i$ is a finite $\Lambda/I$-module. Since $\mathcal{F}/I\mathcal{F}$ has tor amplitude in $[0, 0]$ we conclude that $\overline{M}_i$ is a flat $\Lambda/I$-module. Hence $\overline{M}_i$ is finite projective (Algebra, Lemma \ref{algebra-lemma-finite-projective}). Hence we can find a projector $\overline{p}_i : (\Lambda/I)^{\oplus t} \to (\Lambda/I)^{\oplus t}$ whose image maps isomorphically to $\overline{M}_i$ under the map $\alpha_i$. We can lift $\overline{p}_i$ to a projector $p_i : (\Lambda^\wedge)^{\oplus t} \to (\Lambda^\wedge)^{\oplus t}$\footnote{Proof: by Algebra, Lemma \ref{algebra-lemma-lift-idempotents-noncommutative} we can lift $\overline{p}_i$ to a compatible system of projectors $p_{i, n} : (\Lambda/I^n)^{\oplus t} \to (\Lambda/I^n)^{\oplus t}$ and then we set $p_i = \lim p_{i, n}$ which works because $\Lambda^\wedge = \lim \Lambda/I^n$.}. Then $M_i = \Im(p_i)$ is a finite $I$-adically complete $\Lambda^\wedge$-module with $M_i/IM_i = \overline{M}_i$. Over $U_i$ consider the maps $$ \underline{M_i}^\wedge \to (\underline{\Lambda}^\wedge)^{\oplus t} \to \mathcal{F}|_{U_i} $$ By construction the composition induces an isomorphism modulo $I$. The source and target are derived complete, hence so are the cokernel $\mathcal{Q}$ and the kernel $\mathcal{K}$. We have $\mathcal{Q}/I\mathcal{Q} = 0$ by construction hence $\mathcal{Q}$ is zero by Lemma \ref{lemma-derived-complete-zero}. Then $$ 0 \to \mathcal{K}/I\mathcal{K} \to \underline{\overline{M}_i} \to \mathcal{F}/I\mathcal{F} \to 0 $$ is exact by the vanishing of $\text{Tor}_1$ see at the start of this paragraph; also use that $\underline{\Lambda}^\wedge/I\overline{\Lambda}^\wedge$ by Modules on Sites, Lemma \ref{sites-modules-lemma-completion-flat} to see that $\underline{M_i}^\wedge/I\underline{M_i}^\wedge = \underline{\overline{M}_i}$. Hence $\mathcal{K}/I\mathcal{K} = 0$ by construction and we conclude that $\mathcal{K} = 0$ as before. This proves the result in case $a = b$. \medskip\noindent If $b > a$, then we lift the map $\rho$ to a map $$ \tilde \rho : (\underline{\Lambda}^\wedge)^{\oplus t}[-b] \longrightarrow K $$ in $D(X_\proetale, \Lambda)$. This is possible as we can think of $K$ as a complex of $\underline{\Lambda}^\wedge$-modules by discussion in the introduction to Section \ref{section-derived-completion-noetherian} and because $X_\proetale$ is weakly contractible hence there is no obstruction to lifting the elements $\rho(e_s) \in H^0(X, \mathcal{F})$ to elements of $H^b(X, K)$. Fitting $\tilde \rho$ into a distinguished triangle $$ (\underline{\Lambda}^\wedge)^{\oplus t}[-b] \to K \to L \to (\underline{\Lambda}^\wedge)^{\oplus t}[-b + 1] $$ we see that $L$ is an object of $D_{cons}(X, \Lambda)$ such that $L \otimes_\Lambda^\mathbf{L} \underline{\Lambda/I}$ has tor amplitude contained in $[a, b - 1]$ (details omitted). By induction we can describe $L$ locally as stated in the lemma, say $L$ is isomorphic to $$ (\underline{M^a})^\wedge \to \ldots \to (\underline{M^{b - 1}})^\wedge $$ The map $L \to (\underline{\Lambda}^\wedge)^{\oplus t}[-b + 1]$ corresponds to a map $(\underline{M^{b - 1}})^\wedge \to (\underline{\Lambda}^\wedge)^{\oplus t}$ which allows us to extend the complex by one. The corresponding complex is isomorphic to $K$ in the derived category by the properties of triangulated categories. This finishes the proof.