Quick questions on Proof by contradiction - WJEC A-Level Maths

Euclid's argument: assume there are only finitely many primes $p_1, p_2, \ldots, p_n$. Form $N = p_1 p_2 \cdots p_n + 1$. Then $N$ leaves remainder $1$ when divided by each prime, so no prime in the list divides $N$. Hence either $N$ is itself a new prime, or it has a prime factor not in the list.

State the first line of a proof by contradiction that $\sqrt{5}$ is irrational. [1 mark]

In Euclid's prime proof, why does $N = p_1 \cdots p_n + 1$ have no prime factor in the list? [2 marks]

Prove by contradiction that there is no smallest positive rational number. [3 marks]