Taylor's Theorem/One Variable

Proof using Cauchy Mean Value Theorem
An alternative proof, which holds under milder technical assumptions on the function $f$, can be supplied using the Cauchy Mean Value Theorem.

Proof using Rolle's Theorem directly
Yet another proof for Lagrange Form of the Remainder can be constructed applying Rolle's theorem directly $n$ times; this proof might be easier to visualize geometrically.