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    Real Analysis Continuity

    Homework Statement Prove that if f is uniformly continuous on [a,b] and on [a,c] implies that f is uniformly continuous on [a,c]. Homework Equations The Attempt at a Solution This is my rough idea for a proof, can someone help be say this more formally? Is my thinking even...
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    Real Analysis Limits

    Homework Statement Prove that if f: R->R is an even function, then lim x->0 f(x)=L if and only if lim x->0+ f(x)=L. Homework Equations The Attempt at a Solution So far I have: If f is an even function f(x)=f(-x) for x in domain of f. Then I am trying to apply the limit...
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    Limit involving floor function

    Homework Statement Evaluate lim x-->infinity [x]/x and lim--> -infinity [x]/x. Homework Equations The Attempt at a Solution The think the limits for both of these are 1. I also know that [x] is the largest integer not greater than x. I think that I can use the squeeze theorm...
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    Epsilon Delta Limit Definition

    Homework Statement Prove lim x--> -1 1/(sqrt((x^2)+1) using epsilon, delta definition of a limit Homework Equations The Attempt at a Solution I know that the limit =(sqrt(2))/2 And my proof is like this so far. Let epsilon >0 be given. We need to find delta>0 s.t. if...
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    Number Theory Euclidean Algorithm

    Homework Statement Suppose that u, v ∈ Z and (u,v) = 1. If u | n and v | n, show that uv | n. Show that this is false if (u,v) ≠ 1. Homework Equations a | b if b=ac [b]3. The Attempt at a Solution I understand this putting in numbers for u,v, and n but I don't know how to...
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    Analysis Divergent Sequences

    Homework Statement Prove that the given sequence diverges to infinity. {an} = (-n^4+n^3+n)/(2n+7) Homework Equations Diverges definition The Attempt at a Solution So far I have: Let M>0 and let N= something. I'm having a hard time figuring out what N should equal for the...
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    Advanced Calculus Sequence Convergence

    Homework Statement Prove that the sequence {a_n} converges to A if and only if lim n--->∞ (a_n-A)=0. Homework Equations The Attempt at a Solution It's an if and only if proof, but I'm not sure how to prove it. Please help!
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    Advanced Calculus Sequences

    Homework Statement Determine whether the given limit exists and find their values. Give clear explanations using limit properties. Homework Equations lim n--->∞ (n^2)/n! The Attempt at a Solution I know that the limit is 0, but I don't know how to show it in detailed steps...
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