So the base case that they prove is S(1). And the inductive step is to show: S(1) ->(implies) S(2) -> S(3) -> S(4) -> ....

Or to write it more succinctly, that S(n) -> S(n + 1) assuming S(n) is true.

In this particular problem, the proof that is provided can be used to show S(2) -> S(3), that S(3) -> S(4), etc... are valid and true.

However, the proof could NOT be applied to show that S(1) -> S(2).

So whilst you CAN assume S(2) is true in a proof, the whole inductive chain needs to be attached to a valid base case.

The page gives a better explanation than I could :-)