From Algebra, Chapter $0$ by Aluffi:
In the second line of the proof, he writes assume the statement is known for fewer ideals, but then says we have to prove it for the case of two ideals.
It seems like something is incorrect here.
Isn't the induction hypothesis already assuming that $R \to \big(R/(I_1 \cdots I_{k-1})\big) \times R/I_k$ is surjective?
The proof is completely fine. By induction hypothesis we assume that $R \rightarrow R/I_1 \times \dots \times R/I_{k-1}$ is surjective if the the ideals $I_j$ and $I_l$ are pairwise coprime for all $1 \leq j \neq l \leq k - 1$. Now we consider the map $R \rightarrow R/I_1 \times \dots \times R/I_{k-1} \times R/I_k \cong R/I_1 \cdots I_{k-1} \times R/I_k$. Thus we deal with the two ideals $I_1 \cdots I_{k-1}$ and $I_k$ (which are coprime under the assumption that $I_j$ and $I_l$ are pairwise coprime for all $1 \leq j \neq l \leq k$) and their quotients which means we have to show it for the case of two ideals.