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Upsidealien

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## Homework Statement

Where

:- ∆0 = ∆ is the original triangle ABC.

:- DEF are the midpoints of AB, BC, AC respectively.

:- f1, f2, f3 map the triangular region ABC to the triangular region ADF, DBE and FEC respectively.

:- ∆n+1 = f1(∆n) ∪ f2(∆n) ∪ f3(∆n) for n≥0.

(these are just definitions of Sierpinskis gasket)

1. We need to first sketch ∆1 and ∆2 and prove that ∆n+1 ⊆ ∆n for all n ≥ 0.

We then define S = intersection of ∆n from n=1 to ∞.

2. We then need to prove that S is non-empty and that S = f1(S) ∪ f2(S) ∪ f3(S).

## Homework Equations

Given above.

## The Attempt at a Solution

I attempted 1. by using induction but did not get very far and for 2. I have proved that S ⊇ f1(S) ∪ f2(S) ∪ f3(S) but I am struggling to prove S ⊆ f1(S) ∪ f2(S) ∪ f3(S) Thanks

Tom

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