For all consider the family of bounded functions
We verify for all i.e., the family is pairwise disjoint. Now, choose dense and use that is non denumerable.
I'm studying for comps, and this one has me stumped:
Show that the space of bounded functions under the sup norm is not separable.
I suspect that you'd either have to show an arbitrary dense set is uncountable or that an arbitrary countable set can't be dense, but I don't have any idea how to actually implement those strategies.