Bounded Sequences
A sequence is considered bounded if there exists a real number M such that: $$ | a_n | \le M $$
Any sequence that has a finite limit is necessarily bounded.
Thus, all convergent sequences are bounded.
Note. There are also examples of bounded sequences that are not regular. For instance, the following sequence oscillates between -1 and +1. It neither converges nor diverges, yet it remains bounded. $$ a_n = ( -1 )^n $$
Proof
Suppose a sequence an converges to l:
$$ \lim_{ n \rightarrow \infty } a_n = l $$
Let’s choose ε equal to 1:
$$ \epsilon = 1 $$
By the definition of the limit, there exists a value v such that:
$$ |a_n - l| < 1 \quad \forall n > v $$
Adding |l| to both sides yields:
$$ |(a_n - l) + l| < 1 + |l| $$
Recognizing that an = |an + l - l|, we can write:
$$ |a_n| < 1 + |l| $$
Therefore, we can conclude that:
$$ |a_n| < M $$
where M is defined as the largest value among the absolute values of the terms of the sequence and 1 + |l|:
$$ M = \max \{ |a_1|, |a_2|, \dots , |a_n|, 1 + |l| \} $$
And so on.
