Csdrhrt

The definition of a limit of a complex function goes as follows:- Let a function f be defined at all points z in some deleted neighbourhood of z°. Then the limit of f(z) as z approaches z° is a number w° . That means for each positive number ε , there is a positive number δ such that |f(z)- w°| is less than ε whenever| z-z° | lies between 0 & δ.

Now I am asked to prove

i). limiting value of ( Re z) as z tends to z° equals (Re z°).

ii). limiting value of conjugate of z equals conjugate of z° as z tends to z°.

using definition given above.

I have difficulty as I don't know how to start.

i) Note that as $z to z_0$, $forall epsilon > 0,$ take $delta := epsilon$ such that $forall z: 0 < |z-z_0| < delta, $
$$|mathrm{Re}(z) - mathrm{Re}(z_0)| le sqrt{(mathrm{Re}(z) - mathrm{Re}(z_0))^2 + (mathrm{Im}(z) - mathrm{Im}(z_0))^2} = |z-z_0|<epsilon.$$

$$text{Hence, } lim_{zto z_0} mathrm{Re}(z) = mathrm{Re}(z_0).$$

ii) As $z to z_0$, $forall epsilon > 0,$ take $delta := epsilon$ such that $forall z: 0 < |z-z_0| < delta, $
$$|overline{z}-overline{z_0}| = |overline{z-z_0}| = |z-z_0|<epsilon.$$

$$text{Hence, } lim_{zto z_0} overline{z} = overline{z_0}.$$