Here's why:
* The Sun is incredibly far away: The average distance between the Earth and the Sun is about 93 million miles. One mile is a minuscule fraction of that distance.
* The Sun's energy output is relatively stable: The Sun's energy output varies slightly over time, but these variations are too small to be noticeable with a one-mile shift in distance.
* The Earth's orbit is not a perfect circle: The Earth's orbit is slightly elliptical, meaning it's closer to the Sun at certain points in its journey than others. A one-mile change wouldn't even be noticeable within the natural variations of our orbit.
In short, moving the Sun one mile closer would have no significant impact on Earth.