In the thesis we introduce infinitesimals to present a new approach to classical analysis. We do so by introducing a set of five axioms. The emphasis is on the transfer axiom which helps us transfer theorems from nonstandard analysis to classical analysis. Throughout the thesis we show that the definitions in non-standard analysis are equivalent to those from classical analysis, and use non-standard definitions to prove theorems from classical analysis. Due to the large reach of classical analysis, we limit ourselves to concepts of continuity, limits, differentiation and compactness.
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