1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Sequences and continuous functions

  1. Feb 2, 2012 #1
    1. The problem statement, all variables and given/known data
    a) Let {s[itex]_{n}[/itex]} and {t[itex]_{n}[/itex]} be two sequences converging to s and t. Suppose that s[itex]_{n}[/itex] < (1+[itex]\frac{1}{n}[/itex])t[itex]_{n}[/itex]

    Show that s [itex]\leq[/itex]t.

    b) Let f, g be continuous functions in the interval [a, b]. If f(x)>g(x) for all x[itex]\in[/itex][a, b], then show that there exists a positive real z>1 such that f(x)[itex]\geq[/itex]zg(x) for all x[itex]\in[/itex][a, b].

    2. Relevant equations

    3. The attempt at a solution

    Ok, so I've already done part a. I'm trying to figure out part b. I think my ideas are on the right track, but I'm looking for some help fleshing them out a bit more.

    Argue by contradiction. Suppose there does not exist z such that f(x)[itex]\geq[/itex]zg(x). I'm not sure exactly what this implies in terms of deriving a contradiction; some general guidance would be appreciated it. Also, I'm using the definition that the notion that f(x) is continuous at c is equivalent to the following:

    For every sequence {s[itex]_{n}[/itex]} in the domain of f converging to c, one has

    lim, n-->[itex]\infty[/itex] f(s[itex]_{n}[/itex])=f(c). I somehow want to construct an analogous argument to that in part a, but I'm not sure how the z is similar to the (1+[itex]\frac{1}){n}) term. Can anyone help with this?
  2. jcsd
  3. Feb 2, 2012 #2
    I think this can go easier, define h:

    [tex]f(x)-g(x)=h(x) [/tex]

    because f,g are continuous in [a,b], h is continuous in [a,b] but then h(x) has a minimum >0 in [a,b] call it c thus:

    [tex] f \geq g+c = g \left( 1+\frac{c}{g(x)} \right) [/tex]

    This holds for all x in [a,b] thus:

    [tex] f \geq g+c = g \left( 1+\frac{c}{ ||g(x)||_{\infty} } \right) = zg(x)[/tex]

    with [tex] ||g(x)||_{\infty} =sup_{x \in [a,b]} g(x) [/tex] and [tex] z= 1+\frac{c}{ ||g(x)||_{\infty} }[/tex]

    Something like this should work ( I think the part with the inf term should be replaced by something else).
    Last edited: Feb 2, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook