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

A finite sequence (of rationals, but equally applies to reals) $a_1,a_2,a_3,...,a_n$ is bounded by M iff [...] for all $1\ge i\ge n$.

An infinite sequence $(a_n{)}_{n=1}^{\mathrm{\infty }}$ is bounded by M iff [...] for all $i\ge 1$.

A sequence is said to be bounded iff it is bounded by some $M\ge 0$.