This is a homework and pardon me for the huge gap of my Mathematics knowledge. After thinking and referencing for a few days I came up with something like following, appreciate help to comment whether this is already correct:
By observing the following, it shows that the length of a Bezier curve converges to 1/n of its polygon's length
You almost got it. But you made two mistakes:
(1) In the second line, the control points of the derivative curve are $n\|\Delta C_i\|$, not $\|\Delta C_i\|$.
(2) In going from the second to the third line, you assumed that the norm of a sum of vectors is equal to the sum of their norms. This isn't true, in general. However the norm of the sum is less than or equal to the sum of the norms.
With those two changes, you have what you want.
Your conclusion that the arclength is $\tfrac1n$ times the polygon length is obviously false. Think about what this means geometrically, for $n=2$ or $n=3$.