Stolz-Cesàro Theorem/Corollary

Corollary to Stolz-Cesàro Theorem
Let $\sequence {a_n}$ and $\sequence {b_n}$ be sequences in $\R$ such that $\sequence {b_n}$ is strictly increasing and $\ds \lim_{n \mathop \to \infty} b_n = \infty$.

If:
 * $\ds \lim_{n \mathop \to \infty} \frac {a_n - a_{n - 1} } {b_n - b_{n - 1} } = L \in \R$

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

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:
 * $\ds \sum_{i \mathop = 1}^n x_i = a_n$

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

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

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
 * $\ds \lim_{n \mathop \to \infty} \frac {a_n} {b_n} = \lim_{n \mathop \to \infty} \frac {\sum_{i \mathop = 1}^n x_i} {\sum_{i \mathop = 1}^n y_i} = L$

Hence the result.

Remarks

 * The theorem also holds if $\sequence {b_n}$ is strictly monotone and $\ds \lim_{n \mathop \to \infty} b_n = \pm \infty$.
 * Just as the general Stolz-Cesàro Theorem, the corollary also a version using limit inferior and limit 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.