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 [...] for all $$1 \ge i \ge n$$.
An infinite sequence $$(a_n)_{n=1}^{\infty}$$ is bounded by M iff [...] for all $$i \ge 1$$.
A sequence is said to be bounded iff it is bounded by some $$M \ge 0$$.