No. Induction on n shows that "P(n)" is true for all finite n- it does not extend to infinite values.
I have to prove that if is Hausdorff for each , then is also Hausdorff.
I have proved this for the two dimensional case, and I know if A is finite then it would be true for all of them by induction, but it doesn't say anything about A being finite or infinite. Is this still true by induction if A is infinite?
Take a pair of distinct points . Then for some index , where . Since is Hausdorff, we can find disjoint open subsets and of containing and respectively.
Now let for , and , and similarly take for , and . Then and are disjoint open subsets of containing and respectively, showing that is Hausdorff.