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## Homework Statement

Question: Let ##I = [0,1]##. Suppose ##f## is a continuous mapping of ##I## into ##I##. Prove that ##f(x) = x## for at least one ##x∈I##.

## Homework Equations

Define first(##[A,B]##) = ##A## and second(##[A,B]##) = ##B## where ##[A,B]## is an interval in ##R##.

## The Attempt at a Solution

Proof: let ##M = sup\ f(I)## and ##m = inf\ f(I)##. Assume that ##M ≠ m## and that ##f(x) ≠ x## for all ##x∈ I##.Then ##f## is strictly monotonic and so let ##L_1## = ##[f(m),f(M)]## and define ##L_n ##=## [f(first(L_{n-1}), f(second(L_{n-1}))]## (##n ≥ 2##). Then V=## \bigcap L_n## is non empty. Now let ##y∈V##. Then there exists ##x_1∈ V## such that ##f(x_1) = y##. Then if ##x_1 ≠y##, we have ##f(x_1) ≠ y##, which is a contradiction and so ##x_1 = y##. But this is contradicting our assumption. For the case where ##M = m## it is trivial as the function would be a constant one.