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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).

Tao, Analysis I