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The generalization of De Bruijn sequences to infinite sequences with respectto the order $n$ has been studied iand it was shown that every de Bruijnsequence of order $n$ in at least three symbols can be extended to a de Bruijnsequence of order $n + 1$. Every de Bruijn sequence of order $n$ in two symbolscan not be extended to order $n + 1$, but it can be extended to order $n + 2$.  A natural question to ask is if this theorem is true with respect to thealphabet. That is, we would like to understand if we can extend a De Bruijnsequence of order $n$ over alphabet $k$ into a into a De Bruijn sequence oforder $n$ and alphabet $k+1$. We call a De Bruijn sequence with this propertyan Onion De Bruijn sequence. In this paper we show that the answer to thisquestion is positive. In fact, we prove that the well known Prefer Max DeBruijn sequence has this property, and in fact every sequence with thisproperty behaves like the Prefer max De Bruijn sequence.

On Embedding De Bruijn Sequences by Increasing the Alphabet Size