- #1

jjr

- 51

- 1

## Homework Statement

Prove that [itex] \lim_{z\rightarrow z_0} Re\hspace{1mm}z = Re\hspace{1mm} z_0 [/itex]

## Homework Equations

It is specifically mentioned in the text that the epsilon-delta relation should be used,

[itex] |f(z)-\omega_0| < \epsilon\hspace{3mm}\text{whenever}\hspace{3mm}0<|z-z_0|<\delta [/itex].

Where [itex] \lim_{z\rightarrow z_0}f(z) = \omega_0 [/itex]

Other equations that might be useful are

[itex] |z_1+z_2| \leq |z_1| + |z_2|\hspace{3mm}\text{(Triangle inequality)} [/itex]

and perhaps

[itex] Re\hspace{1mm}z \leq |Re\hspace{1mm} z| \leq |z| [/itex]

## The Attempt at a Solution

Here [itex] \omega_0 = Re\hspace{1mm}z_0[/itex] and [itex] f(z) = Re\hspace{1mm} z [/itex],

so we want to find a delta [itex] |z-z_0| < \delta [/itex] such that [itex] |Re\hspace{1mm}z - Re\hspace{1mm}z_0| < \epsilon [/itex].

I am honestly not sure how to approach this. Any clues would be very helpful.

Thanks,

J