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Thread: definition of a limit

  1. #1
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    definition of a limit

    Suppose $\displaystyle g:[0,\infty)\rightarrow\mathbb{R}$ and $\displaystyle L\in[-\infty,\infty]$. Prove that $\displaystyle lim_{t\rightarrow\infty}g(t)=L \iff$ for every increasing sequence $\displaystyle (a_n)$ in $\displaystyle [0,\infty)$, if $\displaystyle a_n\rightarrow\infty$, then $\displaystyle g(a_n) \rightarrow L$.

    I'm trying to prove [<==] when $\displaystyle L=\infty$ by contradiction. But not sure about the definition:
    $\displaystyle lim_{t\rightarrow\infty}g(t)=\infty$ means that $\displaystyle \forall M>0 $, there is $\displaystyle x\in[0,\infty)$ such that $\displaystyle \forall t>x, g(t)>M.$ Is this the correct definition?
    So if $\displaystyle lim_{t\rightarrow\infty}g(t)\not=\infty$, then there exists $\displaystyle M>0 $ such that $\displaystyle \forall x\in[0,\infty)$ there exists $\displaystyle t>x$ such that $\displaystyle g(t) \leq M.$ Right?
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  2. #2
    Senior Member roninpro's Avatar
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    Hello.

    Your definition seems to be correct.

    Now, you can try for a contradiction/contrapositive by constructing an increasing sequence $\displaystyle \{a_n\}$ which diverges to infinity but $\displaystyle \{g(a_n)\}$ does not.
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