Proposition 1.10 ii), Chinese Remainder Theorem, A&M Introduction to Commutative Algebra

Question

I am working through Introduction of commutative algebra and am having trouble with the following question: (I'll use f instead of the map,since I don't know how to input it.)

Question 1: Why there exist a $x$ such that $f(x)=(1,0,..,0)$?

I can't find a $x$ such that $x+a_1=1$ and $x+a_i=0$ since $x+a_1=1$ implies $x=1$; $x+a_i=0$ implies $x\in \bigcap a_i$

Question 2: Why $f(x)=(1,0,...,0)$ implies $f$ is surjective?

Answer 1

Q1: By assumption $\phi$ is surjective.

Q2: The product is generated by the $n$ elements $$(1,0,\ldots,0), (0,1,0,\ldots,0),\ldots,\ (0,\ldots,0,1).$$