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My text says that the signals emitted at a physical size of horizon $l_{H}(t)$ come to an observer at time $t$ with an infinite redshift. How do I show this mathematically? My approach is as follows:

\begin{equation}l_H(t)=a(t)\eta(t)=a(t)\int_0^t\frac{dt'}{a(t')}=\frac{1}{1+z(t)}\int_0^t(1+z(t'))dt'.\end{equation}

Now I have no idea how to conclude this. Can anyone please help me?

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    $\begingroup$ Typically you integrate the coordinate time from/to the horizon and show the result is infinite $\endgroup$ Commented Jun 20, 2014 at 6:46

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