Counting points within 100km Buffer of country borders using ArcGIS Desktop?

I have a dataset composed of points and projected on a shapefile describing European countries.

My goal is to create a new shapefile, with zones specified 100 kilometers around the borders in-between all the States, since I want to see if there are more points close to the borders, or not.

I thought that Buffer would be a nice way to do it but did not manage to find out how to practically do this. I converted the polygon shapefile once to line, and then used the function Buffer, but there is nothing appearing. I did the same without converting before.