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The presenter switches from numerical examples to an algebraic proof because checking specific cases like $6\times8=48$ and $2\times4=8$ only verifies the statement for those particular pairs. To prove the universal statement that the product of *any* two even numbers is even, a general algebraic argument is required to cover all possible even numbers.

Conditions: The goal is to prove a universal statement about all even numbers.; Numerical examples are used only to build intuition.; Algebra is necessary for a general proof.

The presenter switches from numerical examples to an algebraic proof because checking specific cases like $6\times8=48$ and $2\times4=8$ only verifies the statement for those particular pairs. To prove the universal statement that the product of *any* two even numbers is even, a general algebraic argument is required to cover all possible even numbers.

Conditions: The goal is to prove a universal statement about all even numbers.; Numerical examples are used only to build intuition.; Algebra is necessary for a general proof.

Constructing $x+1$ creates a contradiction because the proof initially assumes that $x$ is the last natural number. However, by adding 1 to $x$, we get $x+1$, which is still a natural number and is strictly larger than $x$.

Conditions: The proof assumes there is a finite amount of $\mathbb{N}$.; $x$ is defined as the last natural number under this assumption.; The arithmetic property used is that if $x \in \mathbb{N}$, then $x+1 \in \mathbb{N}$ and $x+1 > x$.

Constructing $x+1$ creates a contradiction because the proof initially assumes that $x$ is the last natural number. However, by adding 1 to $x$, we get $x+1$, which is still a natural number and is strictly larger than $x$.

Conditions: The proof assumes there is a finite amount of $\mathbb{N}$.; $x$ is defined as the last natural number under this assumption.; The arithmetic property used is that if $x \in \mathbb{N}$, then $x+1 \in \mathbb{N}$ and $x+1 > x$.