The fact that the Hubble radius equals c/H can easily be shown if you assume the universe has the shape of the surface of a 4-sphere, with a current radius of 13.80 billion light-years that is increasing at the speed of light. Using that model, a simple formula for Hubble’s parameter, and, also the Hubble radius, can be easily derived. Since Hubble’s parameter is defined as a VELOCITY divided by a relevant DISTANCE, from the model, since the radius of the 4-sphere is increasing with a velocity of c, it means the velocity at which its circumference is increasing is 2pic when the length of the circumference is 2piR, so H=(2pic)/(2piR), which simplifies to H=c/R. Rearranging, you see that R=c/H, which is what you wanted to ‘prove’. In so far as the model from which this formula is derived is correct, that formula is correct. But about the only way to check the model is to check the predictions of the formula. The observational values of H vary between 68 and 74 km/s/MPc. Plugging in 13800 MLY for the current radius of the 4-sphere (current radius of the 4-sphere equals the age of the universe – 13.80 billion years - times c) one finds:
H=c/R = 299,792,458 m/s /13800 MLY = 21,724 m/s/MLY= 70.85 km/s/MPc
for the 4-sphere’s current surface expansion rate, which is very close to the WMAP probe’s value of 71.0 km/s/MPc, and in the middle of other determinations of H (68 to 74 km/s/Mpc), so the formula, and therefore the model on which the formula was based may be correct.
The formula also shows that the length of the Hubble radius is always the same as the length of the radius of the universe’s 4-sphere, and both are increasing at the speed of light, therefore, both are increasing in length by one light-year per year. (The two are not the same, however. The 4-sphere’s radius is in 4D space, whereas the Hubble radius is in 3D space.) Lastly, R is a physical distance.