Show that the given type of function on a compact metric space has a unique fixed point.
Assume that (X, d) is a compact metric space, and let f: X -> X be a function such that the inequality d(f(x), f(y)) < d(x, y) holds for all distinct elements x, y in X. Show that f has a unique fixed point. See attached file for full problem description.
Real Analysis: Use lower/upper integral to determine Riemann integrability
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Real Analysis: Lipschitz continuous
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Real Analysis - Step Functions:
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Real Analysis - Step Function/Riemann Integral
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Real Analysis - Step Functions
Show that the product of two step functions is a step function. Notes from section of book attached.
Real analysis - pairwise disjoint open intervals
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Real analysis - limit, joint, meet
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Real analysis - set of measure zero
Show that a countable set in R^n is of measure zero. Notes for this section are attached.
Real analysis - set of measure zero
Show that an (n-1)-dimensional face E of an n-dimensional interval is a set of measure zero in R^n. Notes for this section are attached.
