I'm reading a book on ultrafilters and it tells me that showing addition on ultrafilters over the naturals (addition on the set of ultrafilters '$\beta \mathbb{N}$', defined in the standard way) is not commutative is a 'nice exercise'. I am curious to see a proof: however, this result isn't particularly important for what I need ultrafilters for, I don't really want to spend a long time trying to obtain the result.
I have seen that addition is left-continuous, so I suppose the equivalent statement is that addition is not right-continuous (as it would be if $+$ was commutative) though I suspect non-right-continuous and non-commutative immediately follow from one another; could anyone perhaps direct me to a source with a proof that addition is not commutative on $\beta \mathbb{N}$, if they know one? (Or perhaps if the proof is short and it's no more trouble, you could explain the proof yourself - I am happy with either).
Many thanks,