No, it follows from the prime number theorem that, for every $\epsilon>0$ there is $n_0$ such that for every $n>n_0$ there is a prime in the interval $[n,n+\epsilon n]$. In fact the number of such primes is asymptotically $\epsilon n/\log n$ as $n\to\infty$.

Of course much better results are known. For example, Baker-Harman-Pintz (2001) showed that for every $n>n_0$ there is a prime in the interval $[n,n+n^{0.525}]$. Moreover, Carneiro-Milinovich-Soundararajan (2019) proved under the Riemann Hypothesis that for every $n>n_0$ there is a prime in the interval $[n,n+(22/25)\sqrt{n}\log n]$.

No, it follows from the prime number theorem that, for every $\varepsilon>0$ there is $n_0$ such that for every $n>n_0$ there is a prime in the interval $[n,n+\varepsilon n]$. In fact the number of such primes is asymptotically $\varepsilon n/\log n$ as $n\to\infty$.

Of course much better results are known. For example, Baker-Harman-Pintz (2001) showed that for every $n>n_0$ there is a prime in the interval $[n,n+n^{0.525}]$. Moreover, Carneiro-Milinovich-Soundararajan (2019) proved under the Riemann Hypothesis that for every $n>n_0$ there is a prime in the interval $[n,n+(22/25)\sqrt{n}\log n]$.

No, it follows from the prime number theorem that, for every $\epsilon>0$ there is $n_0$ such that for every $n>n_0$ there is a prime in the interval $[n,n+\epsilon n]$. In fact the number of such primes is asymptotically $\epsilon n/\log n$.

Of course much better results are known. For example, Baker-Harman-Pintz showed that for every $n>n_0$ there is a prime in the interval $[n,n+n^{0.525}]$. Moreover, Carneiro-Milinovich-Soundararajan proved under the Riemann Hypothesis that for every $n>n_0$ there is a prime in the interval $[n,n+(22/25)\sqrt{n}\log n]$.

No, it follows from the prime number theorem that, for every $\epsilon>0$ there is $n_0$ such that for every $n>n_0$ there is a prime in the interval $[n,n+\epsilon n]$. In fact the number of such primes is asymptotically $\epsilon n/\log n$ as $n\to\infty$.

Of course much better results are known. For example, Baker-Harman-Pintz (2001) showed that for every $n>n_0$ there is a prime in the interval $[n,n+n^{0.525}]$. Moreover, Carneiro-Milinovich-Soundararajan (2019) proved under the Riemann Hypothesis that for every $n>n_0$ there is a prime in the interval $[n,n+(22/25)\sqrt{n}\log n]$.