Interleave sequence

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

: z_i := left{egin{matrix} x_k & mbox{ if } i=2k mbox{ is even,}\ y_k & mbox{ if } i=2k+1 mbox{ is odd.} end{matrix} ight.

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).


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