Compactness of {0} union {2^-m + 2^-n}

[buzzmath]

I'm trying to show that this set is compact {0} union {2^-m + 2^-n} where n,m are integers and m is less than n

I'm thinking that since it's bounded below by 0 and the set contains it's bound that if I can show that it's closed and bounded above that will be a proof but I'm not really sure how i would do that
or
Say there's an collection of open sets who's union is (-2, 2^-m + 2^-n + 2) that since that's an open cover then and you can have a finite subcover of something like (-1, 2^-m + 2^-n +1) that'll still contain the set therefore the set is compact.

I don't know if either of those are correct or if I'm heading in the right direction. Can anyone help?

 

[LeBrad]

Zero is in the set, so any cover of the set must cover zero. Can you show that all but a finite number of points in the set are covered by this cover on {0}?