You say f is a function. From the context I assume the domain is Nat or f is a sequence. Mathematically speaking, you should form the partial sums of f ( sum f from 1 to n), which is a monotonic sequence of nats. Now proof by contradiction: if your conclusion doesn’t hold, for any m, you can find n > m, such that f n = 1. Then the partial sum goes up by 1 at n. As m is arbitrary, f is not summable.

However I don’t know what is the best way to carry out the above proof in hol as I am not familiar with the relevant libraries yet. Haitao On Friday, March 1, 2019, Chun Tian (binghe) <binghe.l...@gmail.com> wrote: > Hi, > > I'm blocked at the following goal: > > I have a function f returning either 0 or 1. Now I know the infinite > sum of f is finite, i.e. > > suminf f < PosInf (or `summable f` speaking reals) > > How can I prove the set of {x | f x = 1} is finite, or after certain > index m all the rest f(n) are zeros? > > ∃m. ∀n. m ≤ n ⇒ (f n = 0) > > If I use CCONTR_TAC (proof by contradiction), I can easily derive the > following 2 assumptions using results I established in my previous > similar questions: > > INFINITE N > ∀n. n ∈ N ⇒ (f n = 1) > > but still I've no idea how to derive a contradiction with `suminf f < > PosInf` by proving `suminf f = PosInf`... > > Thanks, > > Chun Tian > >

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