Non-nonstandard analysis: real infinitestimals

Infinitesimals have long baffled mathematicians since these are easy to manipulate algebraically yet difficult to understand. A method for the non-nonstandard analysis of infinitesimals is discussed. The method is based on existing calculus texts and allows the natural construction of infinitesimals. The manipulation of sequences of real numbers without using the Axion of Choice and Transfer Principle is described.