An alternative proof is simply obtained with the rearrangement inequality.

Torricelli tried several alternative proofs, attempting to prove it's surface area to actually was finite - all of which failed.

For an alternative proof, consider matrix properties.

An alternative proof starts from the limit definition of :

The result was first proved by Otto Hölder in 1887; several alternative proofs have subsequently been found.

An alternative proof goes as follows, using its own diagram:

Brauer's induction theorem was proved in 1946, and there are now many alternative proofs.

An alternative proof uses the completeness theorem, which is otherwise reduced to a marginal role in most of modern model theory.

For an alternative proof using Cauchy's convergence test, see Alternating series.

We follow the treatment in Bressoud's book; for an alternative combinatorial proof see the paper by Zeilberger.