"Every composite positive integer has at least one prime factor less than the square root of the integer."
Proof by contradiction:
If $p_1, p_2, …, p_n$ are prime factors of $x$ greater than $\sqrt{x}$. Then the following holds for $ā$ $i,jāā$ $i,j<n$;
$$p_i > \sqrt{x}$$
$$p_j > \sqrt{x}$$
$$p_ip_j > x$$
$$CONTRADICTION!$$
The question I have now is that; "Is it possible for a positive integer to have more than one prime factor greater than its square root?"
Considering two prime factors of $x$, $p_1$ and $p_2$ greater than $\sqrt{x}$ using the same reasoning above we have
$$p_1p_2 > x$$
$$CONTRADICTION!$$
This means a positive composite integer can only have one (but may not have any) prime factor greater than its square root.
This is only a proof of what I think is right but I still have a weird feeling that I'm missing something because I've not come across a theorem like this before. The first one I wrote above is a very popular one, and if what the answer I gave is valid then how come it is not popular(unless it isn't true).
Best Answer
What you have shown is this:
If an integer has two prime factors, then at least one of them does not exceed the square root of the integer. So the answer to your question is "No."