a) Show that the alternating harmonic series ∑n=1∞n(−1)n+1 has a rearrangement such that for every interval (x,y), there exists a partial sum sn of the rearranged series such that sn∈(x,y).
b) Show that the rearrangement you found does not converge. See Example 2.6.4.
c) Show that for every x∈R, there exists a subsequence of partial sums {snk}k=1∞ of your rearrangement such that limk→∞snk=x.