Definition:Subsequence

Definition
Let $\left \langle {x_n} \right \rangle$ be a sequence in a set $S$.

Let $\left \langle {n_r} \right \rangle$ be a strictly increasing sequence in $\N$.

Then the sequence $\left \langle {x_{n_r}} \right \rangle$ is called a subsequence of $\left \langle {x_n} \right \rangle$.

Examples
Let $\left \langle {n_r} \right \rangle$ be the sequence in $\N$ defined such that $n_r = r+1$.

Then $\left \langle {x_{n_r}} \right \rangle = \left \langle {x_{r+1}} \right \rangle = x_2, x_3, x_4, \ldots$.

Let $\left \langle {n_r} \right \rangle$ be defined such that $n_r = 2r$.

Then $\left \langle {x_{n_r}} \right \rangle = \left \langle {x_{2r}} \right \rangle = x_2, x_4, x_6, \ldots$.

Let $\left \langle {n_r} \right \rangle$ be defined such that $n_r = 2^r$.

Then $\left \langle {x_{n_r}} \right \rangle = \left \langle {x_{2^r}} \right \rangle = x_2, x_4, x_8, \ldots$.

Note
The constraint that $\left \langle {n_r} \right \rangle$ be strictly increasing is important.

Thus, for example, $x_3, x_1, x_4, x_2, x_9, x_5 \ldots$ is not a subsequence of $\left \langle {x_n} \right \rangle$.