Elementary insight into Bertrand's postulate

Bertrand's postulate states that for any positive integer there is a prime with .

The following is not a rigorous proof but rather something I have thought of which makes the theorem intuitively evident, unlike the rigorous proofs which exist (all of which I have seen are very elegant, but tackle the problem in a roundabout fashion).

Suppose we are given an interval of integers. Call the proportion of integers in which are multiples of . Then

is trivial. In perticular

can be thought of as the proportion of all integers which are

*not* divisible by any of the primes less than

. If we have an interval containing

numbers, we can expect approximately

of those numbers to be divisible by none of the primes less than

.

Note that we have the ridiculously bad bound

so that in perticular any interval containing

numbers should certainly be expected to contain an integer not divisible by any prime less than

, since

. In perticular, when this is applied to the interval

, we see that it should very reasonably contain such an integer, which, in this case, would mean a prime.