How do I show that $B \to A \oplus C$ where $b \mapsto (p(b), j(b))$ is surjective?

Question

From Hatcher's Algebraic Topology:

How do I show that $B \to A \oplus C$ where $b \mapsto (p(b), j(b))$ is surjective?

I can show that this map is well-defined and we use the fact that $\ker (j) = Im(i)$ to show that it is injective. However, I can't seem to show that it is surjective.

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I know that $i(a) \mapsto (p(i(a), j(i(a))=(a,0)$. So, I need to figure out what maps to $(0,c)$.

Answer 1

Let $b \in j^{-1}(c)$. So, $j(b)=c$. Thus, $b-ip(b) \mapsto (p(b-ip(b)), j(b-ip(b)))= (0, c).$

So, $i(a)+b-ip(b) \mapsto (a,c)$.