Defining Positive integers.. While reading Calculus by Apostol I  found the set of positive integers defined as "Set of Real numbers that belong to every Inductive set"...

The question is "Why we don't define the set of 1,1+1,1+1+1,...
I mean all such numbers as the set of positive integers?"

He has said that this will not define the set completely and so we use  "Inductive sets"   

but can someone please explain it a bit more so it becomes actually clear to use inductive sets to define them..
The problem is, what does "all such numbers" actually mean?  You can write down any single number like $1+1$ or $1+1+1+1+1$ easily, but how can you say exactly what the set of all of them is?  You need to define what it means for a number to be "a sum of some number of copies of $1$", but how can you do that when you don't know what "number" (which in the context of this phrase would mean a positive integer number) means?  In order to rigorously prove things about the natural numbers, you will need a completely precise definition of what they are.  And it is quite difficult to say precisely what you mean by "all such numbers" in a non-circular way.  So you need to find a more precise way of defining the natural numbers, and that is the purpose of the definition using inductive sets.
For discrete mathematics the natural numbers (which is practically the same as positive integers without zero) is typically defined using 0 and the successor function S (the successor function being the same +1) $ x \in \mathbb{N} \iff x = 0 \lor \exists y \in \mathbb{N} : S(y)=x $.

Which is effectively the same definition you used so the claim that there exist positive integers that are not covered by that definition runs counter to the standard use of the term "Positive Integer".