# Courses in Math - Theorems - Stolz's theorem

## Stolz's Theorem.

This theorem is very useful when one can find a limit of expressions xn/yn that are of ∞/∞ type.

Let yn → +∞. Moreover, starting from some point yn is increasing with n i. e.

yn+1 > yn

then

lim xn/yn = lim [xn - xn-1]/[yn - yn-1]

if only the limit on the right hand side exists.

##### Proof.

Let us assume that the rhs limit equals a finite number L. Then for ε > 0 there is such an index N that for n > N is

|[xn - xn-1]/[yn - yn-1] - L| < ε/2

and

L - ε/2 < [xn - xn-1]/[yn - yn-1] < L + ε/2.

It means that for any n > N all fractions

[xN+1 - xN]/[yN+1 - yN], [xN+2 - xN+1]/[yN+2 - yN+1], … [xn-1 - xn-2]/[yn-1 - yn-2], [xn - xn-1]/[yn - yn-1]

are between L - ε/2 and L + ε/2. Because from some point

yn+1 > yn

among them is also the fraction (obtained as a sum of all nominators and denominators)

[xn - xN]/[yn - yN]

and for n > N is

|[xn - xN]/[yn - yN] - L| < ε/2.

Let us use the following relation

xn/yn - L = [xN - LyN]/yn + (1 - yN/yn)([xn - xN]/[yn - yN] - L)

|xn/yn - L| ≤ |[xN - LyN]/yn| + |[xn - xN]/[yn - yN] - L|.

The first expression on rhs is less than ε/2 for n > N' > N because yn → ∞. The second one is less than ε/2 as well. Thus one has

|xn/yn - L| < ε.

what should be shown. On the other hand, if the limit

[xn - xn-1]/[yn - yn-1] = +∞

then

xn - xn-1 > yn - yn-1

it can be only if

xn → +∞.

Moreover, terms of sequence {xn} increases with n.