Stolz-Cesàro Theorem/Corollary

Corollary to Stolz-Cesàro Theorem
Let $\left({a_n}\right)$ and $\left({b_n}\right)$ be sequences such that $\left({b_n}\right)$ is strictly increasing and $\displaystyle \lim_{n \to \infty} b_n = \infty$.

If:
 * $\displaystyle \lim\limits_{n \to \infty} \frac{a_n - a_{n-1}}{b_n - b_{n-1}} = L \in \R$

then also:
 * $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = L$

Proof
Define the following sequences:
 * $x_1 = a_1$, $x_n = a_n - a_{n-1}$
 * $y_1 = b_1$, $y_n = b_n - b_{n-1}$

It follows that:
 * $\displaystyle \sum_{i \mathop = 1}^{n} x_i = a_n$

and:
 * $\displaystyle \sum_{i \mathop = 1}^{n} y_i = b_n$

From above follows:
 * $\displaystyle \lim_{n \to \infty} \frac{x_n}{y_n} = \lim_{n \to \infty} \frac{a_n - a_{n-1}}{b_n - b_{n-1}} = L$

From the definition of divergent sequences there exists $N \in \N$ such that
 * $b_n$ is positive for all $n > N$

From the general Stolz-Cesàro Theorem follows that
 * $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = \lim_{n \to \infty} \frac{\sum_{i \mathop = 1}^{n} x_i}{\sum_{i \mathop = 1}^{n} y_i} = L$

Hence the result.

Remarks

 * The theorem also holds if $\left({b_n}\right)$ is strictly monotone and $\displaystyle \lim_{n \to \infty} b_n = \pm \infty$.
 * Just as the general Stolz-Cesàro Theorem, the corollary also a version using limits inferior and superior. In that case the limit $L$ can be a real number or $\pm \infty$.
 * This form of Stolz-Cesàro Theorem is most commonly found in books.
 * The theorem can be considered as a special case of L'Hopital's Rule for sequences.