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Let $\varepsilon$ be an arbitrary small positive number. Can we prove that there exist an $n\in Z$ such that the interval $[2^n,(1+\varepsilon)2^n]$ contain a prime number?

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Yes, by any of several results (starting with Hoheisel) which assert that for sufficiently large $x$ one has at least one prime in $(x,x+x^\theta)$, where $\theta$ is given in advance and is some real number less than one. The current record is like 0.535 for $\theta$. –  The Masked Avenger Jan 31 at 7:42
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Actually all you need is the prime number theorem, which guarantees primes in $(x,(1+\epsilon)x)$ if $x$ is large. So for all large $n$ there are primes in your interval. –  Lucia Jan 31 at 9:46

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