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I know that $a_n=1+\frac1n$ converges to $1$.

How do you prove that $b_n=\left(1+\frac{1}{n}\right)^c$ converges to $1$ where $c\in \mathbb{N}$ is a constant(unfortunaley originally I wrote $c\in\mathbb{R}$ which led to all the comments), and why doesn't the same proof work for $c_n=\left(1+\frac{1}{n}\right)^n$ which converge to $e$.

Please try to prove it using elementary tools (such as the definition of limit, limit arithmetic ...).

Thank you very much.

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    There are many proofs, from fundamentals, or using more machinery. For machinery, use the fact that if $f(x)=x^c$, then $f$ is continuous at any positive $x$. That proof cannot be imitated in the $c_n$ case.2012-05-02
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    @André Nicolas thanks, however I still haven't learned about continuity.2012-05-02
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    Geez, thanks for letting me know you didn't know about continuity; just wasted twenty minutes writing an answer you can't understand yet.2012-05-02
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    @Arturo Magidin your answer is terrific! I'll understand it later in my course(probably within few weeks) it's still a great answer and I try to understand what I can from what I currently know.2012-05-02
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    @Anonymous: Well, the fact is you don't know whether it is terrific or not! No reflection on you (this is a perfectly fine question to ask even before knowing about continuity), but given that it talks in terms of things you don't know, for all you know it's just as confusing as your current status. The reason we ask people to provide explicit context is precisely so that answers can be given at an adequate level. Most US students don't learn sequences until after they've learned continuity, but it's certainly possible to do it in other orders.2012-05-02
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    P.S. How do you define $a^c$ for $c$ an *arbitrary real* before you know about continuity? Do you already know about exponentials and logarithms? What do you know about them?2012-05-02
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    A question similar to the last part of your question have been asked recently: [Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1?](http://math.stackexchange.com/questions/136784/why-lim-limits-n-to-infty-left1-frac1n-rightn-doesnt-evaluate-to).2012-05-02
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    Notice that, when you define $x^c$, you are essentially defining it as a continuous function w.r.t $x$. You can escape this when $c \in \mathbb{Z}$ and, to some extent, when $c \in \mathbb{Q}$. But $2^\pi$ is defined as a limit, so that $x \mapsto x^c$ turns out to be continuous by assumption. In a more polite way, you can use logs and exp's, but you need to use their continuity.2012-05-02
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    I learned about exponentials but not about logarithms yet. We started the course with sequences and we have just started talking about functions.2012-05-02
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    @Anonymous: so you *don't* really know what something like $2^{\sqrt{2}}$ *means*; so you can't really talk about "arbitrary exponents"; so you can't really set up the problem with "$c\in\mathbb{R}$" (which requires the notion of continuity to really define reasonably).2012-05-02
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    @Arturo Magidin Oops! I re-edit the question! I actually do know what $2^{\sqrt{2}}$, but I wanted to ask about when $c\in\mathbb{N}$ and not $c\in\mathbb{R}$ :-(2012-05-02
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    It's easy enough to explain the first using limit laws, and to give a heuristic you shouldn't expect the second limit to "just be equal to $1$". Proving that the second limit actually *isn't* with the tools you seem to have at your disposal is much, much harder.2012-05-02

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