In order to prove, using the [itex]\epsilon-N[/itex] definition, that the sequence(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\left\{\frac{n^2}{n^2+4}\right\} = \left\{\frac{1}{1+\frac{4}{n^2}}\right\}[/tex]

converges towards 1, I did the following: The sequence converges towards 1 if

[tex]\forall \epsilon>0, \exists N\in\mathbb{R} \ \mbox{such that} \ \forall n\in\mathbb{N}, n>N \Longrightarrow \left|\frac{1}{1+\frac{4}{n^2}}-1\right|<\epsilon[/tex]

We have that

[tex]\left|\frac{1}{1+\frac{4}{n^2}}-1\right|<\left|\frac{1}{1+\frac{4}{n^2}}\right|[/tex]

so if we can find the N for the creature on the right side of the inequality, it will also be true for the one on the left side. And then we solve and I spare you the following steps because my question is that if my line of reasoning is correct so far, then proving that there exist an N for the right member proves that there exists an N for

[tex]\left|\frac{1}{1+\frac{4}{n^2}}-a\right|[/tex]

where a in any positive real. So it doesn't REALLY show that the limit is 1 because should I try to prove that the limit is "a, any positive real", I would arrive to the same answer! All it shows is that the sequence converges. Right?

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# Homework Help: Isn't this weird? (convergence of a sequence)

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