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  1. M

    Real Analysis Continuity

    Thanks. What changes should I make to make it more formal?
  2. M

    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...
  3. M

    Real Analysis Limits

    lim x->0- f(x)=L implies that there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(x)-Ll<epsilon provided 0<a-x<delta. I'm am getting confused with all these definitions though, can you help me organize the argument using the definitions?
  4. M

    Real Analysis Limits

    So lim x->0+ f(x)=L implies there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(x)-Ll<epsilon provided 0<x-a<delta. Then lim x->0+ f(-x)=L implies that there exists a real number L s.t. epsilon>0 there exists delta>0 s.t. lf(-x)-Ll<epsilon provided 0<x-a<delta. I have...
  5. M

    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...
  6. M

    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...
  7. M

    Epsilon Delta Limit Definition

    When I try to rationalize, I get l(-2x^2)/[4((x^2)+1)^(1/2)+2(2)^(1/2)*((x^2)+1)]l What did I do wrong?
  8. M

    Epsilon Delta Limit Definition

    I don't know how to simplify it at all. My thought was to maybe to get common denominator or and then multiply by the conjugate, but I don't know if this is correct. I got l[2-(2)^(1/2) *((x^2)+1)^(1/2)]/(2((x^2)+1)^(1/2)l. I'm sorry, it's hard to type it here, does any of that make sense?
  9. M

    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...
  10. M

    Number Theory Euclidean Algorithm

    So then au divides n and vb divides n?
  11. M

    Number Theory Euclidean Algorithm

    Linear combination of u and v are equal to the gcd correct? And the gcd divides u and v I believe. I need help organizing all these ideas.
  12. M

    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...
  13. M

    Analysis Divergent Sequences

    N>(2M)^1/3 Is that correct?
  14. M

    Analysis Divergent Sequences

    I'm sorry, I meant (-n^3)/2. So do I set (-n^3)/2 > M and solve for n?
  15. M

    Analysis Divergent Sequences

    I get (-n^3/2). Then I"m not sure how to solve for the N that I need.
  16. M

    Analysis Divergent Sequences

    I'm sorry, I still don't understand how to write the proof for this divergent sequence. I have a similar problem done correctly I believe. Can someone help me write a proof like the following one, but for this sequence above?? Here is a similar lim, I think I have correctly proved...
  17. M

    Analysis Divergent Sequences

    Yes, I understand that it will diverge to negative infinity. I still need help understanding how to get the N for the proof. Can you help?
  18. M

    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...
  19. M

    Advanced Calculus Sequence Convergence

    I'm not sure. I don't know if I have it written correctly. I feel like I'm working in circles.
  20. M

    Advanced Calculus Sequences

    Yes, I am able to use the product theorem. Thanks for the help!
  21. M

    Advanced Calculus Sequence Convergence

    I'm not sure how to write the reverse. Would I start with: lim n--->infinity(a_n -A)=0. So given epsilon>0, there exists N>0 s.t. la_n-Al<epsilon for all n>N. I don't know if this is correct, and I don't know where to go after that.
  22. M

    Advanced Calculus Sequences

    The limit of the first is 1, and the limit of the next is 0? Correct? Then can I simply multiply 1(0)=0. Will that be enough explanation?
  23. M

    Advanced Calculus Sequence Convergence

    I'm not good at writing proofs. So far I have: Let {an} converge to A. Given epsilon>0, there exists N>0 s.t. lan-Al<epsilon for all n>N. So l((a_n)-A)l<epsilon for all n>n. Thus, we can write lim n--->infinity (a_n-A)=0. Then, I'm not sure how to prove the statement's converse. Can...
  24. M

    Advanced Calculus Sequences

    I have already canceled the factor of n, and I am stuck at the next step. I have n/(n-1)(n-2)! Any suggestions what to do next? How do I proof that =0?
  25. M

    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!
  26. M

    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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