Using my computer, I found that in the interval $[1, N]$ the probability of finding a prime number between $x-\ln(x)$ and $x+\ln(x)$ is greater than constant $c$ where $N=10^2, 10^3,...,10^{9}$, $x$ is the positive integer number in $[1, N]$ and $c > 0.8$.

Question: Does the result hold for $N \to +\infty $?