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 $|a_i|\le M$ for all $1\ge i\ge n$.

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

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

Note how the definition is using only a sequence beginning at index 1 (the sequences of form $(a_n{)}_{n=1}^{\mathrm{\infty }}$ as opposed to the more general form $(a_n{)}_{n=m}^{\mathrm{\infty }}$ for some integer m). Tao mentions earlier in the text that the beginning index is irrelevant.

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Source

Tao, Analysis I