Definition:Subsequence

Definition
Let $\sequence {x_n}$ be a sequence in a set $S$.

Let $\sequence {n_r}$ be a strictly increasing sequence in $\N$.

Then the composition $\sequence {x_{n_r} }$ is called a subsequence of $\sequence {x_n}$.

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

Then:
 * $\sequence {x_{n_r} } = \sequence {x_{r + 1} } = x_2, x_3, x_4, \ldots$

Let $\sequence {n_r}$ be defined such that $n_r = 2 r$.

Then:
 * $\sequence {x_{n_r} } = \sequence {x_{2 r} } = x_2, x_4, x_6, \ldots$

Let $\sequence {n_r}$ be defined such that $n_r = 2^r$.

Then:
 * $\sequence {x_{n_r} } = \sequence {x_{2^r} } = x_2, x_4, x_8, \ldots$

Warning
The constraint that $\sequence {n_r}$ 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 $\sequence {x_n}$.

Also see

 * Definition:Sequence
 * Definition:Subsequential Limit