Problem in understanding the mathematical induction Suppose we have a subset of the set of natural numbers. This set includes 100  numbers that is the first 99 numbers is even and the last number is odd. now, induction can be said that the first number is even(first number mod 2 = 0)  and number n + 1 is even, and so on. consequently all  numbers are even.this is wrong because last number is odd.why is the wrong answer?
 A: It doesn't work since you don't have the following for any number $n$ of your subset of $\mathbb{N}$:
\begin{equation}
n \text{ is even} \Rightarrow n+1 \text{ is even}.
\end{equation}
You clarify this by taking the 99th number of your subset to see that the statement is false.
A: The "inductive" step is to say that if something is true for $n$ then it is true for $n+1$. It's implicit here that this inductive step can performed for ALL $n$. In your example, the inductive step is that if the $n$'th number is even, then so is the $n+1$'th number. The proof of this fact for your example comes from the fact that if $n<99$ then $n+1\leq 99$ which means that the $n+1$'th number must be even by the assumptions of the problem. This does not hold for $n=99$ so the inductive step fails there. 
A: Induction proofs consist in having a case that is true (as you did), then proof through sound facts that if one generic case is true then the next case is true as well (here is where your example fails).
Let's use the following example: every natural number (counting number) is less than 100.
We start with 0. Run the test: $0<100?$ yes! OK we now know that there's at least one case for which this is true. Let's assume that it is true for $n$. Is this true for $n+1$?
Your argument was that I look at $1,2,3,\dots$ and they so seem to satisfy the test. But this is not a sound proof. What do we know about sums an inequalities? There isn't any rule that states
$$a<b \Rightarrow a+1<b$$
Nor anything of the sort that we can take advantage of. Therefore we can't complete our proof with induction. That doesn't mean the  statementstatement is false, just that we haven't been able to prove it.
To show it is false, it suffices to find a counter example, and we have plenty of those.
