Decimal expansion

A decimal representation of a non-negative real number r is an expression in the form of a series, traditionally written as a sum

where a₀ is a nonnegative integer, and a₁, a₂, ... are integers satisfying 0 ≤ a_i ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits a_i are assumed to be 0. Some authors forbid decimal representations with a trailing infinite sequence of "9"s. This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

That is to say, a₀ is the integer part of r, not necessarily between 0 and 9, and a₁, a₂, a₃, ... are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.

Assume ${\displaystyle x\geq 0}$. Then for every integer ${\displaystyle n\geq 1}$ there is a finite decimal ${\displaystyle r_{n}=a_{0}.a_{1}a_{2}\cdots a_{n}}$ such that

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