Definition:Subsequence/Proper

Definition
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 $\N$.