1. Let A ⊆ R be a set, let f, g: A ?R be functions and let k ∈ R. Suppose that f and g are uniformly continuous. Prove that f + g are uniformly continuous.
2. Let [a, b] ⊆R be a closed bounded interval, and let f : [a, b] ?[a, b] be a function. Suppose that f is continuous. Prove that there is some c ∈ [a, b] such that f(c) = c. The number c is call a fixed point of f.