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Math Help - stating true sentences

  1. #1
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    stating true sentences

    Please, tell me which of these are true (and why):

    1. Every unbounded sequence is divergent.
    2. Every convergent sequence is bounded.
    3. A decreasing sequence is convergent if and only if it is bounded above.
    4. The arcsin function is increasing an concave downward on the interval <-1,0>.
    5. If a function is continuous on the interval I, then it is integrable over I.
    6. Differentiability is the sufficient but not necessary condition for integrability.

    And I cannot find what the frontier is.
    Let P be a frontier point of the set A. Then
    [A] P is in A, [B] P is the member of the closure of A, [C] P is in A', [D] every neighborhood of P contains at least one element of A and at least one element of A'

    Thanks in advance!
    Last edited by marcopolon2; May 30th 2008 at 02:15 AM. Reason: adding question about frontier
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  2. #2
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    Quote Originally Posted by marcopolon2 View Post
    1. Every unbounded sequence is divergent.
    If a sequence is convergent then it is bounded. Now take the contrapositive statement.
    2. Every convergent sequence is bounded.
    Just answered above.
    3. A decreasing sequence is convergent if and only if it is bounded above.
    Take a_n = -n.
    5. If a function is continuous on the interval I, then it is integrable over I.
    Yes.
    6. Differentiability is the sufficient but not necessary condition for integrability.
    Yes.
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