Want to know how to prove the product of finite infinitesimal or infinitesimal

Want to know how to prove the product of finite infinitesimal or infinitesimal

Infinitesimal is equivalent to bounded, then according to the limit operation rule, the multiplication of finite infinitesimal is equal to the multiplication of finite bounded numbers, which is always less than t = m (1) * m (2) * *M (n), (n is a constant), and t is also a constant, so the multiplication of finite infinitesimals is bounded, that is, the product of finite infinitesimals is infinitesimal

Higher numbers: the operational properties of equivalent infinitesimal
Who knows the operation properties of the equivalent infinitesimal? For example, how to calculate the addition, subtraction, multiplication and division of two equivalent infinitesimals? Are there any other properties? Today, I review this place, and I don't understand how to use the equivalent infinitesimal to calculate the limit. Who can tell me

Finite infinitesimal addition, subtraction, multiplication or infinitesimal
The product of infinitesimal and bounded function or infinitesimal
Infinitesimal divided by a function whose limit is not zero or infinitesimal
A factor of the product can be replaced by an equivalent infinitesimal, and a part of the sum cannot be replaced
For example: X → 0, neither TaNx nor SiNx in TaNx SiNx can be replaced by X, but after simplifying TaNx SiNx = TaNx (1-cosx), both TaNx and 1-cosx can be replaced