Two definitions of Euler characteristics In third chapter of "Differential Topology" book by Guillemin and Pollack ,Euler's characteristic is defined as I(∆,∆) and in the end of chapter, an intuitive proof is given to show how this is same as one calculated by using triangulations. They defined vector field on triangulated manifold which has source on vertex ,sink on face and saddle on edge .Is there any rigorous as well as generalized ( to n dim.)  proof of this theorem by using tools from book or other fairly basic tools ?
Yes, the "usual proof" involves some (actually, quite a bit of) hand-waving. As an alternative, take a look at a careful write-up of a proof given by Jonathan Libgober, written as a part of his REU: 

Euler characteristc, Poincare-Hopf theorem and applications. 

With all the details, the proof takes about 16 pages, too long to be reproduced in my answer. An outline is that one first develops Morse theory which connects gradient vector fields of Morse functions and cell decompositions of manifolds. Once the machinery is there, it is fairly easy to see that the index of such a gradient vector field on a manifold $M$ equals $\chi(M)$. Lastly, one argues that any two vector nondegenerate fields  have the same index (for this, Guillemin and Pollack suffice). 

Lastly, Peter May (at University of Chicago) had done an amazing job with his undergraduate students writing useful REU projects, such as the one quoted above.
There is nothing non-rigorous about that proof. It works in arbitrary dimensions once you know that manifolds are triangulable. You need some proof like this because the usual definition of Euler char is in terms of triangulations...