Can anyone give me a example of a proper metric space $(X,d)$ such that $(X,\bar d)$, where $\bar d$ is the induced length metric, is not proper.
I have a example but I am not sure if it is right. Ex-: $X\subset R^2$(plane) $X=[0,1]\times {0}\cup [0,1]\times {1}\cup {\frac{1}{n}}\times[0,1]$.This example can be found in BH book. Please comment.
The set $X=[0,1]\times\{0\}\cup\{0\}\times[0,1]\cup\{\frac{1}{n};n=1,2,\dots\}\times[0,1]$ works. This planar set $X$ is a compact subset of a Euclidean space and therefore proper. (This is not what you wrote but something similar. The rough idea is to have a comb where the spikes accumulate.)
The diameter of $X$ w.r.t. the length metric $\bar d$ is 3 (draw a figure!), so to show that $(X,\bar d)$ is not proper it suffices to show that it is not compact. To this end, consider the following cover: $U_n=\{\frac1n\}\times(\frac13,1]$ for $n\in\{1,2,3,\dots\}$, $U_0=\{0\}\times(\frac13,1]$ and $U_{-1}=X\cap([0,1]\times[0,\frac23))$. These sets are an infinite open cover of $(X,\bar d)$ which has no proper subcover, so the space is not compact.
Note that the covering sets are not open in $(X,d)$ and there is no contradiction with its compactness.