Definition:Subsequence

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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 $\sequence {x_n}$.


Proper Subsequence

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

A proper subsequence $\sequence {x_{n_r} }$ of $\sequence {x_n}$ is a subsequence of $\sequence {x_n}$ which is not equal to $\sequence {x_n}$.


That is, in which there exist terms of $\sequence {x_n}$ which do not exist in $\sequence {x_{n_r} }$.

That is, in which the terms of $\sequence {n_r}$ form a proper subset of $\N$.


Examples

Example 1

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

Then:

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


Example 2

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

Then:

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


Example 3

Let $\sequence {n_r}$ be defined such that $n_r = 3 r + 5$.

Then:

$\sequence {x_{n_r} } = \sequence {x_{3 r + 5} } = x_8, x_{11}, x_{14}, x_{17}, \ldots$


Example 4

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

Then:

$\sequence {x_{n_r} } = \sequence {x_{2^r} } = x_2, x_4, x_8, x_{16}, \ldots$


Warning

In the definition of a subsequence, 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

  • Results about subsequences can be found here.


Sources

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