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Let $f_{n}\left( x\right) =\dfrac {x^{n}} {1+x^{n}}$, $x \in \left[ 0,2\right]$

Show that $f_{n}$ converges pointwise on $[0,2]$.

I know that the function converges to $0$ for $0\leq x < 1$, converges to $1/2$ for $x=1$ and converges to $1$ for $x>1$, but I need help showing the definition of pointwise convergence given $\varepsilon>0$.

Thanks!

  • 2
    you can directly follow your remark, note that to show $f_n$ converges pointwise on $[0,2]$, you only deal with a fixed x, so you can just seperate the cases as you mentioned above.2012-12-14
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    you'd better use $x\in [0,2]$.2012-12-14
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    Take it on a case by case basis. If you noted that for fixed $x$ with $0\le x<1$, you have $0\le f_n(x)\le x^n$, could you then show $f_n(x)\rightarrow 0$? For $1\le x\le 2$ you might start by noting $f_n(x)={1\over \textstyle{1\over x^n}+1}$.2012-12-14
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    You have already proved your result. Is the issue that you need to give an epsilon delta proof (i.e. a more detailed proof) and don't know how to do that for pointwise convergence?2012-12-14

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