Project 1 Define inductively by x1 and xn+1 = xn +1/xn. Show that is increasing

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Project 1

Define inductively by x1 and xn+1 = xn +1/xn. Show that is increasing but diverges. Define inductively by x1 = a, a epsilon R, and xn+1 = x2/n. For what value(s) of a odes converges, and to what? Give an example of a family of open intervals whose intersection is the closed interval [0,1]. Suppose f: R rightarrow R and for V x, y epsilon R, | f{x) f(y)| |x - y|2. Show that f is constant function. Prove that if f is a differentiable on a nonempty interval f, and f'(x) is never zero for x epsilon I, then f must be one-to-one on I.

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