I don't really know how to do this in the complex plane, is it the same as doing it with the real numbers, my answers are just guesses???
(b) Does the sequence (z_n), with z_n = 1/n + (1/ n)i converge to 0 in (C; d)?
z_n=1/n(1+i) which converges to 0 I guess... since 1/n converges to 0.
(c) Does the sequence (z_n), with z_n = 1/n + i converge to i in (C; d)?
1/n converges to 0,so z_n=0+i=i.
(d) Characterise(increasing or decreasing) the convergent sequences in (C; d).