(I assume the graph of f is compact; otherwise, the Hausdorff distance isn’t defined, and there seem to be counter-examples to the claim of the exercise.)
Since each fn is a continuous function from T to itself, it has a fixed point xn by Ex10. Then (xn)n∈N is a sequence of points in a compact space and thus has a limit point x∗.
Let Gf be the graph of f. Assume for a contradiction that x∗∉f(x∗). Then, (x∗,x∗)∉Gf. Since Gf is a compact subspace of the Hausdorff space Rn, it is also closed. Let O⊊T be an open set around (x∗,x∗) disjoint from Gf. Then, we find an ϵ∈R+ such that B((x∗,x∗),ϵ)⊆O. (This uses that T is convex, otherwise the ball would exist in Rn but could fall out of T and hence O.)
Choose N⊆N infinite such that (xn)n∈N is entirely contained in B(x∗,ϵ2). Note that (xn,xn)∈Gfn by construction. Write D for the Hausdorff distance, then D(Gf,Gfn)≥d(Gf,(xn,xn))≥ϵ2∀n∈N. This contradicts the fact that (D(Gf,Gfn))n∈N converges to ⟶0, hence x∗∈f(x∗).
Ex11
(I assume the graph of f is compact; otherwise, the Hausdorff distance isn’t defined, and there seem to be counter-examples to the claim of the exercise.)
Since each fn is a continuous function from T to itself, it has a fixed point xn by Ex10. Then (xn)n∈N is a sequence of points in a compact space and thus has a limit point x∗.
Let Gf be the graph of f. Assume for a contradiction that x∗∉f(x∗). Then, (x∗,x∗)∉Gf. Since Gf is a compact subspace of the Hausdorff space Rn, it is also closed. Let O⊊T be an open set around (x∗,x∗) disjoint from Gf. Then, we find an ϵ∈R+ such that B((x∗,x∗),ϵ)⊆O. (This uses that T is convex, otherwise the ball would exist in Rn but could fall out of T and hence O.)
Choose N⊆N infinite such that (xn)n∈N is entirely contained in B(x∗,ϵ2). Note that (xn,xn)∈Gfn by construction. Write D for the Hausdorff distance, then D(Gf,Gfn)≥d(Gf,(xn,xn))≥ϵ2∀n∈N. This contradicts the fact that (D(Gf,Gfn))n∈N converges to ⟶0, hence x∗∈f(x∗).