Kronecker's lemma

The Kronecker lemma is of limits in mathematics. It is named after the German mathematician Leopold Kronecker.

Lemma

Let be a sequence of real numbers.

Be a monotone, unbounded sequence of positive real numbers.

If converges, as follows.

Conclusion

The above lemma simplifies the setting for all the following statement:

Let be a sequence of real numbers.

If converges, as follows.

Application

The Kronecker lemma can be used to prove the strong law of large numbers.