Closed unit ball of an infinite-dimensional Banach space is not compact

You may want to review the definition and properties of quotients of a normed space.

1. Because $B / A_1$ is a normed space, and it is not the zero space unless $A_1 = B$, which is not the case here ($B$ is infinite-dimensional and $A_1$ is one-dimensional). So it contains at least one nonzero element (remember elements of $B / A_1$ are cosets), and you can multiply this by an appropriate scalar to obtain something of norm $1/2$.

2. If you take a quotient of a normed space by a non-closed subspace, in general you only get a seminormed space (the resulting "norm" may fail to be positive definite). Since we want $B / A_2$ to actually be a normed space (see step 1), we need $A_2$ to be closed.

3. It doesn't matter. It is also true that "the closed unit ball of an infinite-dimensional normed space is not compact".