Math and science::Analysis::Tao::05. The real numbers
Bounded sequences
Let \( M >0 \) be a rational.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 \( (a_n)_{n=1}^{\infty} \) 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 \( (a_n)_{n=1}^{\infty} \) as opposed to the more general form \( (a_n)_{n=m}^{\infty} \) for some integer m). Tao mentions earlier in the text that the beginning index is irrelevant.
Two propositions/lemmas that follow:
1. All finite sequences are bounded.
2. All Cauchy sequences are bounded. (proof: exercise 5.1.1).