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Engineering Calculus Notes 353

Engineering Calculus Notes 353 - convergent sequence s i...

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3.6. EXTREMA 341 (a) Any close interval [ a,b ] in R ; (b) any half-closed interval of the form [ a, ) or ( −∞ ,b ]; (c) any level set L ( g,c ) of a continuous function g ; (d) any set defined by weak inequalities like { x R 3 | g ( x ) c } or { x R 3 | g ( x ) c } ; 10. Prove Remark 3.6.9 : (a) For any set S R 3 , S int S ∂S. (b) The boundary ∂S of any set is closed. (c) S is closed precisely if it contains its boundary points: S closed ∂S S. (d) S R 3 is closed precisely if its complement R 3 \ S := { x R 3 | x / S } is open . Challenge problems: 11. (a) Show that any set consisting of a
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Unformatted text preview: convergent sequence s i together with its limit is a closed set; (b) Show that any set consisting of a (not necessarily convergent) sequence together with all of its accumulation points is a closed set. 12. Prove that if α,β > 0 satisfy 1 α + 1 β = 1 then for all x,y ≥ xy ≤ 1 α x α + 1 β y β as follows: (a) The inequality is clear for xy = 0, so we can assume xy n = 0....
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