Math and science::Analysis::Tao::05. The real numbers

# Sequences bounded away from zero

A sequence \( (a_n)_{n=1}^{\infty} \) is said to be bounded away from zero iff there exists a rational \( c > 0 \) such that \( |a_k| \ge c \) for all integers \( k \ge 1 \).A sequence \( (a_n)_{n=1}^{\infty} \) is said to be positively bounded away from zero iff there exists a rational \( c > 0 \) such that \( a_k \ge c \) for all integers \( k \ge 1 \). (In other words, the sequence is comprised entirely of positive rationals.)

A sequence \( (a_n)_{n=1}^{\infty} \) is said to be negatively bounded away from zero iff there exists a rational \( c > 0 \) such that \( a_k \le -c \) for all integers \( k \ge 1 \). (In other words, the sequence is comprised entirely of negative rationals.)

These properties are used, among other things, to define reciprocation for the reals, ordering of reals and by extension the Archimedean property.