there’s is a typo in my previous post. I wanted to ask “…, why f must *not* be 
summable?” (i.e. divergent)

On a second thought, your approach could work for proving the partial sum is 
unbounded: for any real number e > 0, I can first use
Archimedean property (simple version) of reals to find an integer N such that e 
< N, then I repeat N times your approach, first let the partial sum >= 1, then 
>= 2, … finally >= N, by induction, then the whole proof finished.  However to 
implement this idea it’s not easy to me. Still want to know better proofs.


> Il giorno 01 mar 2019, alle ore 20:54, Chun Tian (binghe) 
> <> ha scritto:
> Hi Haitao,
> thanks, and yes, f is a sequence of reals.  I'm following a similar path
> (proof by contradiction), but I don't understand the last step: for any
> m, the partial sum of f goes up by 1 at n > m, why f must be summable? I
> think for every monotonic sequence such properties hold.
> --Chun
> Il 01/03/19 17:24, Haitao Zhang ha scritto:
>> 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) <
>> <>> 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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