Definition:Subsequence

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 $$\left \langle {x_n} \right \rangle$$.