Cantor's Theorem (Strong Version)/Proof 2

Proof
The proof proceeds by induction.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * There is no surjection from $S$ onto $\map {\PP^n} S$.

Basis for the Induction
$\map P 1$ is Cantor's Theorem.

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * There is no surjection from $S$ onto $\map {\PP^k} S$.

Then we need to show:
 * There is no surjection from $S$ onto $\map {\PP^{k + 1} } S$.