In the sequence limit section of mathematical analysis, there is a very fundamental "Cauchy’s Proposition":
If \lim_{n\to\infty} x_n=a, then \lim_{n\to\infty}\frac{x_1+x_2+\dots+x_n}{n}=a
This article discusses this proposition, as well as some similar problems.
Proof of Cauchy’s Proposition
The proof of Cauchy’s Proposition is not difficult. It only requires the definition of limit convergence. Since \lim_{n\to\infty} x_n=a, for any given \varepsilon > 0, there exists a sufficiently large N such that for all n > N: \left|x_n - a\right| < \varepsilon/2\quad(\forall n > N)
Then, for a sufficiently large n, we have: \begin{aligned} &\left|\frac{x_1+x_2+\dots+x_n}{n}-a\right|\\ =&\left|\frac{(x_1-a)+(x_2-a)+\dots+(x_n-a)}{n}\right|\\ \leq &\left|\frac{(x_1-a)+(x_2-a)+\dots+(x_N-a)}{n}\right|\\ &\quad+\left|\frac{(x_{N+1}-a)}{n}\right|+\left|\frac{(x_{N+2}-a)}{n}\right|+\dots+\left|\frac{(x_{n}-a)}{n}\right|\\ < & \left|\frac{(x_1-a)+(x_2-a)+\dots+(x_N-a)}{n}\right|+\frac{n-N}{n}\frac{\varepsilon}{2}\\ < & \left|\frac{(x_1-a)+(x_2-a)+\dots+(x_N-a)}{n}\right|+\frac{\varepsilon}{2}\\ \end{aligned}
By choosing a sufficiently large M > N > 0, as long as n > M, we have: \left|\frac{(x_1-a)+(x_2-a)+\dots+(x_N-a)}{n}\right| < \frac{\varepsilon}{2} Thus: \left|\frac{x_1+x_2+\dots+x_n}{n}-a\right| < \varepsilon By the arbitrariness of \varepsilon, we conclude: \lim_{n\to\infty}\frac{x_1+x_2+\dots+x_n}{n}=a
In fact, the intuitive meaning of Cauchy’s Proposition is quite obvious. It simply tells us that if a sequence is convergent, it becomes increasingly "flat." When taking the average, even if there are large fluctuations at the beginning, they can be smoothed out by a sufficiently large n.
A Variation
Now, I invite the reader to try this problem:
Given \lim_{n\to\infty} x_n=a, prove that: \lim_{n\to\infty}\frac{x_1+2 x_2+\dots+n x_n}{n^2}=\frac{1}{2}a
Please, do not be in such a hurry to start writing. Stare at it, stare at it intently, until the problem becomes afraid of you and gives up the answer itself!!
In fact, with Cauchy’s Proposition, this problem does not even need a formal proof; it is obviously true! Why? Why should we only consider the sequence \{x_n\}? Why don’t we consider the following sequence instead? \{y_n\}=\{x_1, x_2, x_2, x_3, x_3, x_3, x_4, x_4, x_4, x_4, x_5, \dots\} Clearly, we also have: \lim_{n\to\infty} y_n = a Furthermore: \frac{1}{n^2}\sum_{k=1}^n k x_k = \frac{1}{n^2}\sum_{k=1}^{\frac{1}{2}n(n+1)} y_k = \frac{\frac{1}{2}n(n+1)}{n^2} \frac{\sum_{k=1}^{\frac{1}{2}n(n+1)} y_k}{\frac{1}{2}n(n+1)} From Cauchy’s Proposition, we immediately know that the limit of the summation term on the far right is a, and the limit of \frac{\frac{1}{2}n(n+1)}{n^2} is clearly \frac{1}{2}. Therefore: \lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^n k x_k = \lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^{\frac{1}{2}n(n+1)} y_k = \frac{1}{2}a So, by changing your perspective, you might see something unexpected. Sometimes, if you just keep staring at it, it becomes obviously true. I believe that being able to understand the same problem from different angles is what it truly means to see through a problem.
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