I am studying [Fary-Milnor Theorem](http://mduchin.math.tufts.edu/UCD/116/milnor.pdf) on total curvature of knots and I am stuck in a proof. He is proving on page 9:



> 
> The Total curvature of a tame knot cannot equal the curvature of its type
> 
> 
> 


k(C) := total curvature of knot C


So by assuming false he takes a knot C which k(C)=k([C])
(where [C] := it's isotopic equivalence class) and gets the inscribed polygon, P,which is a member of [C] (proved existence in theorem 4.2 on page 8) 
he create another polygon in [C], P', which have the property of
k(P') < k(P) <= k(C) = k([C]).
and write "which is impossible".
I cannot see why it is impossible. Can someone help?


[milnor proof](https://i.stack.imgur.com/mzSV7.png)


Thanks,



Fary-Milnor Theorem : Help following a proof on page 9
First, the OP's reference is Milnor's paper:


*Milnor, John W.*, [**On the total curvature of knots**](http://dx.doi.org/10.2307/1969467), Ann. Math. (2) 52, 248-257 (1950). [ZBL0037.38904](https://zbmath.org/?q=an:0037.38904).


Secondly, the total curvature of a type is the inf of the curvatures of tame knots of that isotopy type. Milnor shows (using proposition 1.2 in [the paper](http://people.reed.edu/~ormsbyk/milnor-total-curvature.pdf)), that you can always decrease the curvature slightly by an isotopy, so the inf is never attained.


