In an analysis book i found this proof for the uniqness of a limit of a function.(adsbygoogle = window.adsbygoogle || []).push({});

This is not homework

"It is easy to show that when a limit of a function f(z) exists at a point a,it is unique.To do this ,we suppose that

lim f(z) =l and lim f(z)=m as z----> a (z goes to a).

Then,for any positive number ε,there are positive numbers r,δ such that

[tex]\left| f(z)-l\right|[/tex]< ε whenever 0<[tex]\left|z-a\right|[/tex]< r

and

[tex]\left|f(z)-m\right|[/tex] < ε whenever 0<[tex]\left|z-a \right|[/tex]< δ.

So if 0< [tex]\left|z-a\right|[/tex]< θ ,where θ denotes the smaller of the two Nos r and δ,we find that

[tex]\left|m-l\right|[/tex] = [tex]\left|(f(z)-l)-(f(z)-m)\right|[/tex] =< [tex]\left| f(z)-l \right|[/tex] + [tex]\left|f(z)-m\right|[/tex] < ε+ε =2ε.

But [tex]\left|m-l\right|[/tex] is a nonnegative constant, and ε can be chosen arbitrarily small.

Hence

l-m =0 , or l=m."

is that proof correct??

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# Is that proof correct?

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