Take a sequence of complex numbers, and form two new sequences of real numbers, one from the real parts, and one from the imaginary parts. The complex sequence converges if and only if both the real sequences do.
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).
Increasing sequences.
Okay...so for
(b) 1/n -> 0 and (1/n)i ->0
so does this mean that the sequence converges to 0.
and for
(c) 1/n -> 0 and i->i (since its constant)
so this converges to i.
Is this right do we add the the convergence of the real part to the convergence of the complex part???