How can it be proved that there can't be $k$ consecutive positive integers whose product only has those primes found in $k$ factorial?

I am curious about the nature of the primes in

$$ \frac{m(m+1)(m+2)...(m+k-1)}{k!} $$

when $k < m$ ( means they dont overlap).

We can show that it is always an integer using n choose k formulation or pascals rule or by prime factor counting.

But this last proof got me thinking. How can it be shown that there will be other primes as factors than those in $k!$ ?

Edit: I want to show that there is at least one prime greater than k as factor in that in every case.

I am looking for a non-handwavy yet insightful proof.

Edit2: I want to show that there is at least one ( specific form not needed ) prime greater than k that divides the product of any k consecutive natural numbers. just existence proof. In other words, I want to show that there is always a prime greater than k between k and n/2 or between n-k and n, whenever n is strictly greater than 2k.

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