What is proof that acceleration is directed towards the centre of the circle in Uniform circular motion?

Try swinging a ball on a string in a circle. The only force you're providing is via the string which extends from the ball to your hand (which is in the center), so the force must be towards the center.

Were you looking for some other kind of reasoning?

Showing this mathematically is pretty straight forward. In cartesian coordinates, the equation for the point in motion is:


\vec x = R cos(wt)\vec i\\
\vec y = R sin (wt) \vec j


where w = the rotational velocity, in radians/sec, and the vectors \vec i and \vec j are the unit vectors in the x and y directions, respectively.

Take the second derivative of the position to find the acceleration:


\vec {a _ x} = \frac {d ^2 x} {dt ^2} = -Rw ^ 2 cos(wt) \vec i\\
\vec {a _ y} = \frac {d ^2 y} {dt ^2} = -Rw ^2 sin(wt) \vec j


Hence:

\vec {a _x} = -w^2 \vec x\\
\widevec {a _y} = -w^2 \vec y


Hope this helps.