I asked [this question][1] on math.stackexchange, but no one answered.

Let $(X,\le)$ be a totally ordered set. Regarding it as a category, it has a classifying space $B(X,\le)=|N_\bullet(X,\le)|$. This should be the (possible infinite) simplex with vertices $X$, hence I expect it to be contractible.

However, I was not able to explicitly prove contractibility starting from the definition $$B(X,\le)=(\coprod_{i\in\mathbb{N}_0}N_i(X,\le)\times\Delta^i)/\tilde{} $$ 

Can anyone help me?

  [1]: https://math.stackexchange.com/questions/1261999/classifying-space-infinite-totally-ordered-set-contractible