its about induction and i got a contradiction By induction we can show infinite union of finite disjoint set is finite
we prove it by induction
let us take the @ be the family of infinite disjoint finite sets,then we describe elements of @ be of form A1,A2,A3...so on.
then we prove that union of this infinite number of finite disjoint sets to be finite,
we prove it by induction 
for n=1 we see that it is trivially true since we know that set of all @ is a finite set so for n=1 it is true
for n=2 it will be of form $A_1\cup A_2$ and we know that it is also finite it is a theorem and it is proven 
then for n=3 we can take $A_1\cup A_2\cup A_3$ and we know that $A1\cup A2$ is finite then again doing union with a finite set will generate another finite set.
thus let us assume it is true until a 'n' belongs to N.
then we prove it is true for n+1
we assumed that $A_1\cup A_2\cup A_3\cup ......\cup A_n$ is finite 
then let us take $A_1\cup A_2\cup ......\cup A_n=D$
So $D\cup A_{n+1}$ is finite and we know that.
so the thm is true for all 'n' belongs to $\mathbb N$.
but which we know is false -_-'
 A: Note that at each step of your induction, you are taking the union of $n$ sets. So, you managed to prove by induction that the finite union of finite sets is finite, NOT that an infinite union of finite sets is finite.
A: Writing what you proved and what you think you proved in a more rigorous mathematical language may help you understand the difference:
You proved:

If $A_1,A_2,A_3,\dots$ is an infinite family of disjoint sets, then, for every $n\in\mathbb N$, the set $$\bigcup_{i=1}^n A_i$$
  is finite.

You did claim you proved:

If $A_1,A_2,A_3,\dots$ is an infinite family of disjoint sets, then the set $$\bigcup_{i=1}^\infty A_i$$
  is finite.

This second statement cannot be proven by the induction that proves the first statement, because it is a statement that does not contain the $n\in\mathbb N$ over which you can run the induction. 
The first statement tells you that no matter how many sets you take the finite union of, you will still get a finite set. The second statement would tell you that an infinite union obeys the same rule, which is a different (and, in fact, false) statement alltogether.
