Can every positive real be written as the sum of a subsequence of dot dot dot I answered this thing Infinite sum of prime reciprocals and now wonder what happens if we do not have such a strong condition as Bertrand's postulate. i have been fiddling with this, not sure either way.
Given a sequence $a_1 > a_2 > a_3 \cdots$ of strictly decreasing positive reals such that
$$ a_i \rightarrow 0 \; \; \; \mbox{but} \; \; \sum a_i = \infty,  $$
can every positive real number be expressed as the sum of a subsequence of the $a_i?$ The main thing is that we are not given any upper bound on $a_n / a_{n+1}.$ For the reciprocals of the primes, we had an upper bound of $2.$
Note that this is subtler than the thing about rearranging a strictly alternating conditionally convergent series to get anything you specify. That is a matter of overshooting with positive terms, then undershooting with negative terms, back and forth. This one is a little different. 
I think what I want is a careful proof of this: given two positive real numbers $B<C,$ we can find a finite subsequence of the $a_n$ with sum between $B$ and $C.$

I'd say certainly so. Denote $S_{n,m} = \sum_{i=n}^m a_i$. Fix $L \in \mathbb{R}_+$ the desired limit. Set $R_0 = L$. Given $R_i$ for $i \geq 0$ first pick $N_i$ the least integer with $a_{N_i} < R_i$, and $M_i$ to be the minimal integer $M_i \geq N_i$ with $S_{N_i,M_i} < R_i < S_{N_i,M_i + 1}$. Then set $R_{i+1} = R_i - S_{N_i,M_i}$. Note that $N_{i+1} > M_i$, and that $L = \sum_i S_{N_i,M_i}$. Obviously there's a bit of verification. 
On a philosophical level this is very similar to the alternating case, however here we don't allow overshoot as we have culling rather than negative terms.
