Let $$X_1\xrightarrow{f_1}X_2\xrightarrow{f_2}\dots$$ be a sequence of simplicial sets and maps between them (denote corresponding diagram by $I$) and $$X_1\xrightarrow{f_{N-1}\dots f_1}X_N\xrightarrow{f_{2N-1}\dots f_N}\dots$$ be some cofinal subsequence (I am really interested in one case when we take each $N$-th term, corresponding subdiagram is $I'$).
Is it true that in this case $ \mathrm{hocolim}(I)=\mathrm{hocolim}(I') $ ? It is certainly true for usual colimits, but for mapping telescopes we need to check that comma category $(i\downarrow \mathrm{in})$ is not only connected but contractible for any $i\in I$ (here $\mathrm{in}:I'\to I$ is an inclusion functor). Note that this property does not depend on $X_i$ and $f_i$ ! I've tried to draw simplicies in first few dimensions in nerve and it looks like result should be contractible. This should be a very classic result but I can't find a solid reference.
Let $\alpha:\Bbb N\to \Bbb N$ be the functor given by $\alpha(1)=1,\, \alpha(2)= N,\, \alpha(3)=2N,\cdots$. You want to show that $\alpha$ is homotopy cofinal, i.e. $j\downarrow \alpha$ has contractible classifying space. This follows because each of the overcategories $j\downarrow \alpha$ has an initial object. (Alternatively, it also has a terminal object)
For instance