Given a sequence $a_1, a_2, \ldots, a_n$ a subsequence is $a_{i_1}, a_{i_2}, \ldots a_{i_k}$ is such that $i_j < i_j + 1$ (note this can apply to infinite sequences as well). In other words, it is a subset of the sequence ordered such that the indices are increasing.

Example

For the sequence

$$ 5, 7, 4, -3, 9, 1, 10, 4, 5, 8, 9, 3 $$

the following are all subsequences

$$ 5, 7, 4, -3, 9, 1, 10, 4, 5, 8, 9, 3 $$

$$ -3, 9, 1, 10 $$

$$9, 10, 8$$

however

$$ 9, 10, 1 $$

is not as the indices are not increasing.