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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 $\N$.


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.