I see what you are trying to do though and see the confusion.
By definition, a set B is open iff for every x in B, there is an open set containing x and contained in B.
But consider that point 0 that is in B. There is no open interval (these are the basis elements for the real line in the usual topology) which contains 0 and does not intersect the sequence (in fact there are an infinite number of points of this intersection for every open set containing 0 by definition of ).
If instead 0 were removed like the teachers correction states (call it B') what you were trying to do is spot on.
but the way it stands as he originally stated, 0 is in B, but not in B'.