relativistic approximation made in 2 flavor neutrino oscillation derivation

Reading a 2 flavor neutrino oscillation derivation I saw the following approximation being made regarding energy and momentum: E-p ≃ m²/2E

I can't see how this step is taken, and what is being assumed. I would appreciate it if someone could help me understand this.

The approximation is made on page 3-4 of this pdf: http://www.hep.phy.cam.ac.uk/~thomson/partIIIparticles/handouts/Handout_11_2011.pdf

The same approximation is stated at wikipedia: https://en.wikipedia.org/wiki/Ultrarelativistic_limit

But neither explains how it is done.

In the ultrarelativistic limit it is helpful to define the small quantity $\delta v$, which is defined by $v = c-\delta v$. We can start with $$\frac{1}{2\gamma^2} = \frac{1}{2} \left(1-\frac{v^2}{c^2}\right) = \frac{1}{2}\left(1-\frac{c^2-2c\delta v +\delta v^2}{c^2} \right)\approx \frac{\delta v}{c} = 1-\frac{v}{c}$$ The approximation for $E-p$ is now straighforward using the above result $$E-cp = E\cdot(1-\frac{v}{c}) \approx E \cdot \frac{1}{2\gamma^2} = \frac{m c^2}{2\gamma} = \frac{m^2 c^4}{2\gamma m c^2} = \frac{m^2c^4}{2 E}$$