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A proof which can be accomplished using only real numbers (i.e., real analysis instead of complex analysis; Hoffman 1998, pp. 92-93).

For real, nonnegative terms x_n and real p with 0<p<1, the expression lim_(k->infty)x_0+(x_1+(x_2+(...+(x_k)^p)^p)^p)^p converges iff (x_n)^(p^n) is bounded.

The portion of the complex plane z=x+iy with real part R[z]<0.

A spectrum containing the real numbers larger than Freiman's constant.

The portion of the complex plane z=x+iy with real part R[z]>0.

Even though real arithmetic is uncountable, it possesses a countable "model."

An infinite set, such as the real numbers, which is not countably infinite.

If a real algebraic curve has no singularities except nodes and cusps, bitangents, and inflection points, then n+2tau_2^'+iota^'=m+2delta_2^'+kappa^', where n is the order, ...

"The reals" is a common way of referring to the set of real numbers and is commonly denoted R.