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# 2.3.1 adv. calc - en ⎪an-= 0⎪ ⎪an⎪ =< an ε e...

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Nathan Fitzenreider 2.3.1 ( Suppose a ) . n is a sequence of positive real numbers a) ( If a ) → , n 0 t h ( ) → . en an 0 b) ( If a ) → , n x t h ( ) → . en an x c) Proof. a) Because a → , n 0 we know t h at t h ere exists an N N so t h at ⎪an - < 0⎪ ε b) w h . enever n N Since t h e sequence is positive we can say t h at c) ⎪a n - = 0⎪ ⎪ an⎪ = . an Now c h oose N1 N so t h at d) ⎪a n - < 0⎪ ε 2 a < n ε 2 a < . n ε T h en ⎪an - = 0⎪ ⎪an⎪
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Unformatted text preview: en ⎪an-= 0⎪ ⎪an⎪ = < . an ε ) e f) > Let ε 0 and a → . n x C h ∈ oose N N so t h at ⎪an-x⎪ < x ∙ ε w h ≥ . enever n N g) We can rewrite ⎪a n-x⎪- + as ⎪an x x an⎪ . T h is implies t h , at h) ⎪a n-x⎪ = ⎪an-x⎪⎪ + ≤ x an⎪ ⎪an-x⎪⎪ < . x⎪ ε T h → . us an x i) j) k)...
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