[Math] Prove that for any natural number $n$ there exists a prime number $p$ greater than $n$

Question

Prove that for any natural number n there exists a natural prime number p , such that $ p>n $. How can I prove that ? Thank you.

Answer 1

First thing prove that there are infinite number of primes,you can use the Euclid proof imagine that there are $n$ primes and name the $k$-th prime $p_k$ than the number $t=p_1p_2\cdots p_{n-1}p_n+1$ isn't divisible by any prime $p_k$ where $1\leq k\leq n$ so there is another prime which divides $t$.Now it's easy to show that $p_k> k$ for every $k\in\mathbb{N}$ since the first prime is $p_1=2$ and $p_{k+1}>p_k$