Show that $p_{k}≥2k-1$ for all $k≥2$

Question

Let $p_{k}$ be the $k^{th}$ prime. Show that $p_{k}≥2k-1$ for all $k≥2$. This inequality is true for several values of $k$. For example $p_{2}=3≥2(2)-1=3$ and $p_{3}=5≥2(3)-1=5$.

Answer 1

Hint: No even positive integers are prime except for $2$. Alternatively, as rtybase's comment suggested, you can also use mathematical induction.