Proof of Limit: |Rez - Rez0|<E Whenever 0<|z-z0|<D

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The discussion focuses on proving the limit of the real part of a complex number, specifically demonstrating that |Rez - Rez0| < E whenever 0 < |z - z0| < D, where E and D are real numbers greater than 0. The key takeaway is that the condition |Rez - Rez0| <= |z - z0| is crucial for establishing this limit. It is emphasized that the relationship between E and D is essential, as the implication does not hold universally without specific constraints on these values.

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How do I show |Rez - Rez0|<E whenever 0<|z-z0|<D is true, where E and D are real number greater than 0, and z is obviously a complex number?

In other words, proving that the lim of Rez (as z approaches z0)=Rez0.
 
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As literally written, your first statement is not true and doesn't imply the second one; it depends on what E and D are.

However, since |Rez - Rez0| <= |z-z0| is all you need to know you should be able to work it out.
 
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What you mean is to show that, given any E>0, there exist a D>0 such that |Re(z)- Re(z0)|<E whenever |z-z0|< D. Matt grime's point is that that is very different from saying that |Re(z)- Re(z0)|< E whenever |z-z0|< D for any E and D.

And, as he said, it follows from the fact that |Re(z)-Re(z0)|< |z- z0|.
 

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