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Source: published paper, printed p. 413 (PDF p. 3), Lemma 1. The source explicitly imports these facts from its references [6], [7], [8] and [4]; it does not prove them here.

With $p_k$ the $k$th prime and $p_1=2$,

$$\begin{array}{ll} p_k\ge k\log k,& k\ge2,\\[2pt] p_k\le k(\log k+\log\log k),& k\ge6,\\[2pt] p_k\le k\log p_k,& k\ge4,\\[2pt] p_k\ge k(\log p_k-2),& k\ge5. \end{array}$$

These are exact external interfaces, not four newly reconstructed proofs. The third bound is needed to turn a Chebyshev error into a bound proportional to $k$, and the second converts ranges in $p_k$ into ranges in $\log k$.

The cited sources are J. B. Rosser, The n-th prime is greater than n log n (1939), and Explicit bounds for some functions of prime numbers (1941); J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers (1962); and J.-P. Massias and G. Robin, Bornes effectives pour certaines fonctions concernant les nombres premiers (1996). The allocation to this collection follows Dusart's citation; no claim of independent full inspection of those four papers is made here.