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Math Help - Why does this limit of this integral work?

  1. #1
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    Why does this limit of this integral work?

    "If we set epsilon > 0 and N as a large natural number, and if j, k > N:

    \left| \int_{a}^{b} f_j(x) \, dx \right| \, -\, \left| \int_{a}^{b} f_k(x) \, dx \right| \, \leq \int_{a}^{b} \left| f_j(x) - f_k(x) \right| dx

    OK where exactly is this coming from? This appears as one of the initial statements in a proof about f_j's integral approaching f's integral. But I could not find the theorem that states this. Is it supposed to be self-evident or something?
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  2. #2
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    Re: Why does this limit of this integral work?

    all you've written here is the triangle inequality where $\left|\displaystyle{\int_a^b}g(t)~dt\right|$ is a norm of $g(t)$
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  3. #3
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    Re: Why does this limit of this integral work?

    Quote Originally Posted by romsek View Post
    all you've written here is the triangle inequality where $\left|\displaystyle{\int_a^b}g(t)~dt\right|$ is a norm of $g(t)$
    Ah, OK. The triangle inequality for integrals. Got it.
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  4. #4
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    Re: Why does this limit of this integral work?

    It's a little more complicated than that. Using the form of the triangle inequality |y|-|z| \leq |y-z|,

    \left| \int_{a}^{b} f_j(x) \, dx \right| \, -\, \left| \int_{a}^{b} f_k(x) \, dx \right| \, \leq \left| \int_{a}^{b} f_j(x) \, dx - \int_{a}^{b} f_k(x) \, dx \right|= \left| \int_a^b f_j(x)-f_k(x) \,dx \right|

    And using \left| \int a \right| \leq \int |a| gives

    \left| \int_{a}^{b} f_j(x) \, dx \right| \, -\, \left| \int_{a}^{b} f_k(x) \, dx \right| \leq  \int_a^b |f_j(x)-f_k(x)| \,dx

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