The simplicity of nonstandard analysis

Nonstandard analysis (NSA) makes many mathematical arguments and concepts much easier. As an example, we shall look at some elementary calculus.

In NSA, the real line R is extended to a hyperreal line *R, which contains infinite numbers and infinitesimals. Functions f defined on R are extended to hyperreal functions *f defined on *R. Similarly, any subset A of R is extended to a subset *A of *R. Details can be found in many NSA texts.

Definition: x is infinitesimal iff -a < x < a for all positive reals a. Note that 0 is the only real infinitesimal.

Definition: If a, b are in *R, then a is infinitely close to b iff a-b is infinitesimal.

We give the following propositions without proof:

Proposition: A real function f is continuous at a real number b iff *f(x) is inf. close to f(b) whenever x is inf. close to b.

Proposition: A real function f is uniformly continuous on A, a subset of R iff *f(x) is inf. close to *f(y) for all inf close x, y in *A.

These are much simpler than the usual epsilon-delta versions, and easier to work with. Proofs, and a more detailed discussion, can be found in Lindstrom's paper, An invitation to NSA (in NSA and its applications, CUP 1988), and many other NSA texts. Many other mathematical ideas are similarly simplified by the use of NSA.

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