Interleave sequence

In mathematics, an interleave sequence is obtained by merging together two sequences.

Let $S$ be a set, and let $(x\_i)$ and $(y\_i)$, $i=0,1,2,...,$ be two sequences in $S.$ The "interleave sequence" is defined to be the sequence $x\_0,\; y\_0,\; x\_1,\; y\_1,\; dots.$ Formally, it is the sequence $(z\_i),\; i=0,1,2,...$ given by

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Properties

* The interleave sequence $(z\_i)$ is convergent if and only if the sequences $(x\_i)$ and $(y\_i)$ are convergent and have the same limit.

* Consider two real numbers "a" and "b" greater than zero and smaller than 1. One can interleave the sequences of digits of "a" and "b", which will determine a third number "c", also greater than zero and smaller than 1. In this way one obtains an injection from the square (0, 1)×(0, 1) to the interval (0, 1).