In the master's thesis, we study Bertrand's postulate, which states that for any given natural number $n$, there exists at least one prime number $p$ such that $n < p \leq 2n$. We present in more detail some of the most famous proofs of Bertrand's postulate - Erdős's, Ramanujan's and simplified Ramanujan's proof. Erdős's proof is based on estimation of the binomial coefficient. Ramanujan's proof derives from the first proof of Bertrand's postulate, Chebyshev's proof from 1852. Ramanujan's simplified proof was written by Meher and Ram Murty, who proved the postulate in the same way as Ramanujan, except they avoided using Stirling's formula in their proof. We also describe modifications of Bertrand's postulate and their applications, including Ramanujan's primes that Ramanujan introduces at the end of the proof of Bertrand's postulate.
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