Nope, no proof that higher dimensions actually exist.
Higher dimensions are just math concepts used to solve problems that could not be solved without resorting to them. This would be something like 2D people living in a 2D world finding out they could best solve the area of a 2D circle by solving the cross sectional area of a 3D sphere (one higher dimension than the 2D world). And, by a math trick called projection, we know this is possible.
There are three spatial dimensions (e.g., length, width, height) and one temporal (time). So we live in a 4D universe.
Time, more correctly, the rate of passage of time, can be altered, just like the size of a box can be altered. So time is real, not just a math concept. The rate of passage change is given by t = t0/sqrt(1 - (v/c)^2); where t0 is rest time (normal time as you sit and read this answer) and t is the dilated time (slower rate of passage).
The denominator is the so-called Lorentz Transform. If you look at it carefully, you will see that t --> infinity as v --> c the velocity (v) of the frame where the time is observed approaches light speed (c).
At nearly the speed of light, time on a spaceship, for example, might slow down to take millennia on Earth for one second on the ship to click off. What this means is that when those space cadets returned from their star trek, thousands of years will have passed on Earth. They will have traveled forward in time on Earth while barely aging on the ship. If time were not a real dimension, this so-called time dilation would not happen.
Physics and engineering degrees.