s1 - Math 8601: REAL ANALYSIS. Fall 2010 Homework #1....

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #1. Problems and Solutions. #1. Let F be a compact subset of R n . Show that there are point x 0 , y 0 F , such that diam F := sup {| x - y | : x, y F } = | x 0 - y 0 | . Proof . By definition of sup, there are sequences { x j } , { y j } in F , such that diam F = lim j →∞ | x j - y j | . Since F is compact, there are subsequences x j k x 0 F as k → ∞ , and y j k l y 0 F as l → ∞ . Then diam F = lim j →∞ | x j - y j | = lim l →∞ | x j k l - y j k l | = | x 0 - y 0 | . #2. Let f ( x ) be a real function on a compact set E R 1 . Show that f is continuous on E if and only if its graph Γ = { ( x 1 , x 2 ) R 2 : x 1 E, x 2 = f ( x 1 ) } is a compact set in R 2 . Proof . (I) Let f be continuous on E , and let { z j } be a sequence in Γ. We can write z j = ( x j , f ( x j )), where x j E . Since E is compact, there is a subsequence x j k x 0 E . Since f is continuous, we have f ( x j k ) f ( x 0 ). This means
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s1 - Math 8601: REAL ANALYSIS. Fall 2010 Homework #1....

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