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Fixed Point Theorem and Closed Unit Ball in Euclidean Space

The Brouwer Fixed-Point Theorem

Let denote the closed unit ball in Euclidean space :
.
Any continuous map from onto itself has at least one fixed point, i.e. a point such that .

Proof Suppose has no fixed points, i.e. for .
Define a map , , by letting be the point of intersection of and the ray starting at the point and going through . For see figure below:

We have
with , (1)
Then
(2)
and so is continuous. Could you please explain, in as much detail as possible, how (1) and (2) were derived?

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Fixed Point Theorem and Closed Unit Ball in Euclidean Space are investigated. The solution is detailed and well presented.

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