In countably infinite union of countably infinite sets is countable the proof has been given, but when as a student I attempted the question, I tried using induction ( later to found it to be wrong way of going about that problem) but the reasoning was quite simple.
I.Union of 1,2 countable set is counatble ( obvious )
II.Suppose I is true up some n.
III.For case n+1 we get union of all the sets up to n, which was true by II , so we get back to case of union of two countable sets which is true.
I couldn't see any flows with the above, but it was wrong, I think it was due to the fact that proving something holds for all finite n, is not the same as proving it for the infinite case.
My question is, why proving something for all finite n ( after all any nu,ber that can be picked is finite), is not same as proving the infinite case ? Is there any other example induction failing for infinite clause?
Another question is : Is uncountable ( infinite of course :) union of countable ( finite or infinite) is countable? (The wrong induction method I used can't even be used for this one.)
PS : Modified the title as it was sugeestive of induction failing, where is the case is it is being misused.