Have a look!! Hi all, Well, i am posting the work that i have done in proving that theorem. Like i said it is nothing important, but rather it is important only for me, so if you could have a look at what i have done i would really appreciate it. The theorem that i have tried to prove differently is the theorem upon the limit laws. I already know how to prove limit laws using Cauchy's definition of limits, and also using Heine's definition(that is in terms of sequences), but the way i tried to prove them is using only and only infinitesimals. In other words, i first tried to prove some properties of infinitesimals, as a background information, based solely on the definition of infinitesimals. I also have never seen the properties of infinitesimals proven solely on its definition,but rather based on the properties of limits, so if you could pay heed to that part too i would appreciate it. The main point here is the theorem upon the Limit Laws. If someone of you could spare a few minutes and let me know if i have done anything at all, i would really appreciate it. Thanx in advance P.S. I am going to upload the file in two pieces, since it seems to exceed the capacity of uploading here.