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Unformatted text preview: A ⊆ Λ(OI + (w0 , z0 )). That is, for every ǫ > 0 and every
(x, y ) ∈ A, we must ﬁnd (wj , zj ) with 0 < d((x, y ), (wj , zj )) < ǫ. (x,y) B
(wk ,z k )
(x’,y’) C Figure 2.29: Showing (x, y ) is a limit point of {wi , zi }.
To do this, take m large enough that diam(Aim ...i1 ) < ǫ for all length m
addresses. Let B denote such a region with (x, y ) ∈ B . Within this B , take
C = Ain ...im ...i1 ⊂ Aim ...i1 = B with (x, y ) ∈ C . Applying Cor. 2.4.1, we see
C contains a point (x′ , y ′ ) ∈ A. See Fig. 2.29. Take δ > 0 small enough that
D((x′ , y ′ ), δ ), the disc with center (x′ , y ′ ) and radius δ , satisﬁes D((x′ , y ′ ), δ ) ⊂
C . In the proof of Theorem 2.4.1 we saw there is a point (xj , yj ) as close as we
like to (x′ , y ′ ), hence with
d((xj , yj ), (x′ , y ′ )) < δ/2.
By eq (2.12), for this same δ there is a k large enough that
d((xk , yk ), (wk , zk )) < δ/2.
Also from eq (2.12) it follows that for all k ′ ≥ k ,
d((xk′ , yk′ ), (wk′ , zk′ )) < δ/2.
Then if...
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This document was uploaded on 02/14/2014 for the course MATH 290B at Yale.
 Fall '14
 AmandaFolsom
 Geometry, Fractal Geometry, Limits, The Land

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