Originally Posted by

**johnsomeone** Second point, and this one is more subtle, so sorry if it confuses you. Your inductive step doesn't apply in moving from the n=1 case to the n=2 case. Plug n=1 into your inductive assumptions. What does (a1 a2 a3 ... a(n-1)) look like? It doesn't make sense when n=1, does it? It makes sense when n=2 (then (a1 a2 a3 ... a(n-1)) = (a1)). So even ignoring my first point above, what you've proved, with S(n) being your inductive hypothesis for n, is that:

1) S(1) is true.

2) For all n >= 2, if S(n) is true, then S(n+1) is true.