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Prime numbers - Do they end at a point?

Prime numbers are one of the most speculated topics in mathematics. Mathematicians have pondered about them enough to arrive at a proper conclusion to the above question.

Let us look at the first few prime numbers:
$$2\quad3\quad5\quad7\quad11\quad13\quad17\quad19\quad23\quad29\quad31\quad37\quad41$$ Let us now look at the first few prime numbers after $100000$:
$$100003\quad100019\quad100043\quad100049\quad100057\quad100069\quad100103\\100109\quad100129\quad100151\quad100153\quad100169\quad100183$$ If you look closely, you may notice that the average separation between consecutive prime numbers between $100003$ and $100183$ is larger than that of $2$ and $41$. Naturally, our brain thinks that this separation increases on and on as we go to larger numbers, and at one point the set of prime numbers terminate. But that is not the case. It is proven that there are infinitely many prime numbers.

The Proof:

Euclid had the same question we have today, and he gave a beautiful proof that is quoted even today for proving that the set of prime numbers do not terminate.

Let us assume that there are only a finite number of primes, name that set $P$.
$$P = \lbrace a_1, a_2, a_3,...,a_n \rbrace$$
Consider a number $A$ such that
$$A = a_1\times a_2\times a_3\times a_4\times ...\times a_n + 1$$
Since the prime number list ended at $a_n$, $A$ must not be a prime number. So $A$ must have a factor that belongs to the set $P$.

Let $a_m$ be the factor of $A$ where $1\le m\le n$. But it is clear that when $A$ is divided by $a_m$, it leaves a remainder 1! But this contradicts the fact that factors leave a remainder of 0 when dividing the number.

Thus, the original assumption that there is a finite list of prime numbers is false. Hence it is proved by the method of contradiction that there are infinitely many prime numbers.
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