Section 2.2, exercise 38 in Hatcher's algebraic topology textbook is:
I can show that the final sequence $$\cdots \to E_{n+1} \to B_n \to C_n \oplus D_n \to E_n \to B_{n-1} \to \cdots$$
has the property that composing two maps gives zero, but I'm not sure how to go about showing that its exact. (I.e. I've shown that the image of one map is contained in the kernel of the next, but not that the kernel of one map is contained in the image of the previous map)
How do I go about proving exactness? I feel like it all comes down to the commutativity of the diagram (with the vertical arrows) but I'm not entirely sure how to proceed