Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could perhaps bring up examples of "proofs" which were thought to be groundbreaking but were later proved wrong by subsequent results; on the other hand, you might also consider giving examples of when a theorem was questioned by the mathematical community but was confirmed by independent proofs, causing the theorem to be more widely accepted.

Originally From Comments:

There are plenty of examples. For the first case, perhaps start with this MathOverflow post. For example, that page lists the false result that

A convergent infinite series of continuous functions is continuous. Cauchy gave a proof of this (1821)...Five years later Abel pointed out that certain Fourier series are counterexamples.

A more trivial example is a tougher question. That same page lists the Euler's Formula $V\text{(ertices)}−E\text{(dges)}+F\text{(aces)}=2$ was long thought to hold for all polyhedra, and a proof was given that was initially accepted for many years until a counterexample was found.

As for the second case, immediate examples that come to mind include Cantor's Set Theory (though subsequent proofs were largely due to Cantor himself, with some strong avocation from Hilbert) as well as Godel's Incompleteness Theorem (see the page Rosser's Trick as well as Boolo's Short Proof ). Neither of these examples are fantastic, but I am sure a little research could definitely yield some better examples!

Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could perhaps bring up examples of "proofs" which were thought to be groundbreaking but were later proved wrong by subsequent results; on the other hand, you might also consider giving examples of when a theorem was questioned by the mathematical community but was confirmed by independent proofs, causing the theorem to be more widely accepted.

Originally From Comments:

There are plenty of examples. For the first case, perhaps start with this MathOverflow post. For example, that page lists the false result that

A convergent infinite series of continuous functions is continuous. Cauchy gave a proof of this (1821)...Five years later Abel pointed out that certain Fourier series are counterexamples.

A more trivial example is a tougher question. That same page lists the Euler's Formula $V\text{(ertices)}−E\text{(dges)}+F\text{(aces)}=2$ was long thought to hold for all polyhedra, and a proof was given that was initially accepted for many years until a counterexample was found.

As for the second case, immediate examples that come to mind include Cantor's Set Theory (though subsequent proofs were largely due to Cantor himself, with some strong avocation from Hilbert) as well as Godel's Incompleteness Theorem (see the page Rosser's Trick as well as Boolo's Short Proof ). Neither of these examples are fantastic, but I am sure a little research could definitely yield some better examples!

Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could perhaps bring up examples of "proofs" which were thought to be groundbreaking but were later proved wrong by subsequent results; on the other hand, you might also consider giving examples of when a theorem was questioned by the mathematical community but was confirmed by independent proofs, causing the theorem to be more widely accepted.

Worth noting here is that there is an additional level of certainty in the validity of a statement when you have multiple proofs of said statement. By having multiple ways to solve the same problem, all yielding the same answer, our surity of the statement increases dramatically. In a classroom setting you could perhaps bring up examples of "proofs" which were thought to be groundbreaking but were later proved wrong by subsequent results; on the other hand, you might also consider giving examples of when a theorem was questioned by the mathematical community but was confirmed by independent proofs, causing the theorem to be more widely accepted.

Originally From Comments:

There are plenty of examples. For the first case, perhaps start with this MathOverflow post. For example, that page lists the false result that

A convergent infinite series of continuous functions is continuous. Cauchy gave a proof of this (1821)...Five years later Abel pointed out that certain Fourier series are counterexamples.

A more trivial example is a tougher question. That same page lists the Euler's Formula $V\text{(ertices)}−E\text{(dges)}+F\text{(aces)}=2$ was long thought to hold for all polyhedra, and a proof was given that was initially accepted for many years until a counterexample was found.

As for the second case, immediate examples that come to mind include Cantor's Set Theory (though subsequent proofs were largely due to Cantor himself, with some strong avocation from Hilbert) as well as Godel's Incompleteness Theorem (see the page Rosser's Trick as well as Boolo's Short Proof ). Neither of these examples are fantastic, but I am sure a little research could definitely yield some better examples!