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I have this ugly proof that some sequence converges to something and I really don't like it because it seems too hard for no reason... can anyone help me make this more simple?

Here it is : We are given a sequence $p_n$ with $n p_n \to \lambda > 0$. We show that $$ \left( 1 - p_n \right)^n \to e^{-\lambda}. $$ So my proof begins by noticing that $$ (1-p_n)^n = \left( 1 + \frac{-np_n}n \right)^n $$ and that the family of functions $s_n(x) = \left(1+\frac xn \right)^n$ is equicontinuous at $x$ (I think I can pull some argument stating that the derivative of $s_n(x)$ is always bounded (because it also converges to $e^x$) when considering an interval of arbitrary but fixed length centered at $x$, so that in this interval, all functions $s_n$ can have the same Lipschitz constant, therefore giving equicontinuity, correct me if I'm wrong) so that if I note by $x_n$ the sequence $-n p_n$, $x = -\lambda$ and $s(x) = e^x$, I wish to show that $$ \forall \varepsilon > 0, \quad \exists N \quad s.t. \quad \forall n > N, \quad |s_n(x_n) - s(x)| < \varepsilon. $$ so now, $$ |s_n(x_n) - s(x)| \le |s_n(x_n) - s_n(x_m)| + |s_n(x_m) - s_n(x)| + |s_n(x) - s(x)|. $$ Since $x_n$ is convergent, it is also Cauchy, so that for all $\delta > 0$, there exists an $N_1$ such that for all $n,m > N_1$, $|x_m - x_n| < \delta$ and $|x_m - x| < \delta$. Since the family $\{s_n\}$ is equicontinuous, for all $\varepsilon > 0$, there exists a $\delta > 0$ such that $|x - y| < \delta \Rightarrow |s_n(x) - s_n(y)| < \frac{\varepsilon}3$, so that the first two terms are taken care of by $N_1$ by taking $(x,y) = (x_n, x_m)$ and $(x,y) = (x_m, x)$ respectively. The last term is also bounded by $\varepsilon/3$ when $n > N_2$ because $s_n \to s$ pointwisely. Taking $N = \max \{ N_1, N_2 \}$ we're done.

I feel like my proof is a little too harsh on this easy-looking problem... anyone has a better proof? What I mean by "better" is using less powerful or advanced tools, something simple. I can't seem to find anything better right now.

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Yes, good point. I will –  Patrick Da Silva Nov 8 '11 at 5:04
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3 Answers 3

up vote 10 down vote accepted

Hmmmm... This smells like plain analysis to me: you want to show that $n\log(1-p_n)\to-\lambda$.

First, $\log(1-x)\le -x$ for every $x$ in $(0,1)$ hence $n\log(1-p_n)\le-np_n$. Since $np_n\to\lambda$, $\limsup n\log(1-p_n)\le-\lambda$.

Second, $\log(1-x)\ge-x-x^2$ for every $x$ in $(0,\frac12)$. For every $n$ large enough, $p_n\le\frac12$ hence $n\log(1-p_n)\ge-np_n-np_n^2$. Since $np_n^2\to0$, $\liminf n\log(1-p_n)\ge-\lambda$.

You are done.

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Thank you, now this is sweet. Didn't think about using logarithms, which make obviously things more simple. –  Patrick Da Silva Jun 9 '11 at 15:28
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This could even be done by a(n admittedly very strong) calculus student.

Step 1: For any real number $l$, one has $\lim_{n \rightarrow \infty} (1-\frac{l}{n})^n = e^{-l}$. (Take logarithms and apply L'Hopital's Rule.)

Step 2: Let $M > \lambda$, so that for sufficiently large $n$, $(1-p_n)^n \geq (1-\frac{M}{n})^n$. By Step 1, for any $\epsilon > 0$ and all sufficiently large $n$ we have

$(1-p_n)^n \geq e^{-M} - \epsilon$.

Step 3: Similarly, let $m < \lambda$ and arguing as above for any $\epsilon > 0$ and all sufficiently large $n$ we have

$(1-p_n)^n \leq e^{-m} + \epsilon$.

The result follows!

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Good one. I just noticed how stuck I was looking at the functions instead of thinking about elementary analysis/calculus... –  Patrick Da Silva Jun 9 '11 at 20:12
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It is also quite natural to attack the $n$th power directly using the binomial theorem: $$\begin{align*}(1-p_n)^n&=\sum_{j\geq 0}{n\choose j}(-p_n)^j\\\\&= \sum_{j\geq 0}\left[n(n-1)\cdots(n-j+1)\over n^j\right](-np_n)^j/j!\\\\ &\to \sum_{j\geq 0}(-\lambda)^j/j!\\\\&=\exp(-\lambda).\end{align*} $$ The convergence is obvious termwise, but you need the Dominated Convergence Theorem (say), to justify convergence of the infinite sums.

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I didn't need help on that part, and we weren't allowed to use dominated convergence to do that since we're in a course where it is not even told. Thanks for that though. –  Patrick Da Silva Jul 21 '11 at 15:10
    
@Patrick It's true that turning this into a real proof would be more work than it is worth, since we already have Pete and Didier's nice proofs. I hesistated a long time before posting, but in the end decided that it might be useful for somebody. One of things I like about this approach is that you don't need to know that the limit is $\exp(-\lambda)$ ahead of time. –  Byron Schmuland Jul 21 '11 at 17:13
    
It is useful, even for me. I know the dominated convergence theorem and I like your proof. Don't hesitate to post such answers ; people post here to have everyone's opinions and all types of answers, nice ones included! That's why I +1'ed. –  Patrick Da Silva Jul 22 '11 at 6:40
    
I wish I could +2 this one for its simplicity and quality. But Didier's is what I asked for =S –  Patrick Da Silva Nov 8 '11 at 5:11
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