limit and infinite ordinals: same thing? I'd like to clarify my understanding re:  limit and infinite ordinals.  This post says:  An initial infinite ordinal is a limit ordinal.  
Is it true the other way around?  That is  if I have a limit ordinal must that also be an infinite initial ordinal?
This would seem to be true because for a limit ordinal x, if I have any 'y' less than x, then I have a 'w' such that   y < w < x.     
And the definition of infinite initial ordinal 'x' is:  For all B < x,  we also have B dominated by (of lesser cardinality than) x.  I understand that this means there is no way to have an onto map from B to x.   If x is a limit ordinal, then for all y < x, there is no way to make an onto map from y to x.  This is true because for whatever supposedly onto function f from B to x you give me, I can always take each image of f (in x), and double that image to produce an ordinal in x which is not mapped to by f.
Does this make sense ?
Thanks in advance for confirming! (or showing false, perhaps with a counter-example?)

No. $2\omega$ is a limit ordinal and infinite, but not initial as it is still countable.
