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Assume that there exists m≥1, so:

${S}^{m}=S∘S∘··∘S $,where the length is m

is a contraction.

1) Show that S has a unique fixed point

2) Show that for $m=2$ we can say that $S=cos:[0,\frac{π}{2}]→[\frac{π}{2}]$

**Definition:**

Let X = (X,d) be a metric space.

A map $S: X → X$ is contraction if there exists a number $0≤β≤1$ so:

$d(S_x,S_y )≤βd(x,y)$ for all $x,y∈X$

**[1]**

A fixed point for S is a $x∈X$ with $ T_x=x$

1) Assume that x and y are fixed points for S.

**[1]**implies that $d(x,y)=d(S_x,S_y )≤βd(x,y)$ [2]

Since$ β≤1$ and $d(x,y)≥0$ is [2] only true when d(x,y)=0, which implies that x = y.

Therefore S has a unique fixed point.

2) I'm not sure about this part. Can someone help me with this part, please.