Question 2.45: Given limn→∞ zn = l. Prove that limn→∞ Re{zn} = Re{lg} and l...

Let z_n=x_n+i y_n and l=l_1+i l_2, where x_n, y_n, and l_1, l_2 are the real and imaginary parts of z_n and l, respectively.
By hypothesis, given any \epsilon>0 we can find N such that \left|z_n-l\right|<\epsilon for n>N, i.e.,

or

From this, it necessarily follows that

i.e., \lim _{n \rightarrow \infty} x_n=l_1 and \lim _{n \rightarrow \infty} y_n=l_2, as required.