Kronecker's lemma
Encyclopedia
In mathematics
, Kronecker's lemma (see, e.g.) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers
. The lemma is named after the German
mathematician
Leopold Kronecker
.
is an infinite sequence of real numbers such that
exists and is finite, then we have for
and 
that
denote the partial sums of the xs. Using summation by parts,
Pick any ε > 0. Now choose N so that
is ε-close to s for k > N. This can be done as the sequence 
converges to s. Then the right hand side is:
Now, let n go to infinity. The first term goes to s, which cancels with the third term. The second term goes to zero (as the sum is a fixed value). Since the b sequence is increasing, the last term is bounded by
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Kronecker's lemma (see, e.g.) is a result about the relationship between convergence of infinite sums and convergence of sequences. The lemma is often used in the proofs of theorems concerning sums of independent random variables such as the strong Law of large numbers
Law of large numbers
In probability theory, the law of large numbers is a theorem that describes the result of performing the same experiment a large number of times...
. The lemma is named after the German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Leopold Kronecker
Leopold Kronecker
Leopold Kronecker was a German mathematician who worked on number theory and algebra.He criticized Cantor's work on set theory, and was quoted by as having said, "God made integers; all else is the work of man"...
.
The lemma
If is an infinite sequence of real numbers such thatexists and is finite, then we have for
and thatProof
Let denote the partial sums of the xs. Using summation by parts,
Pick any ε > 0. Now choose N so that
is ε-close to s for k > N. This can be done as the sequence converges to s. Then the right hand side is:
Now, let n go to infinity. The first term goes to s, which cancels with the third term. The second term goes to zero (as the sum is a fixed value). Since the b sequence is increasing, the last term is bounded by
.