  The generalization of De Bruijn sequences to infinite sequences with respect
to the order $n$ has been studied iand it was shown that every de Bruijn
sequence of order $n$ in at least three symbols can be extended to a de Bruijn
sequence of order $n + 1$. Every de Bruijn sequence of order $n$ in two symbols
can 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 the
alphabet. That is, we would like to understand if we can extend a De Bruijn
sequence of order $n$ over alphabet $k$ into a into a De Bruijn sequence of
order $n$ and alphabet $k+1$. We call a De Bruijn sequence with this property
an Onion De Bruijn sequence. In this paper we show that the answer to this
question is positive. In fact, we prove that the well known Prefer Max De
Bruijn sequence has this property, and in fact every sequence with this
property behaves like the Prefer max De Bruijn sequence.
