I think that in particular Erez Lapid has done a nice job with these two slides
http://www.math.clemson.edu/~jimlb/ConferenceTalks/ColumbiaWorkshop2006/lapid1.pdf
http://www.math.clemson.edu/~jimlb/ConferenceTalks/ColumbiaWorkshop2006/lapid2.pdf
Have a look in particular on page 10 in the first slide session for Bernstein's prinicple, and a proof of it is on page 11. He has also states it in for $SL(2)$ on page 9, which is always helpful for me before seeing a general statement. The second slides focus on the higher rank situation.
By the way, the whole side is great: http://www.math.clemson.edu/~jimlb/coursenotes.html
Perhaps a remark about Eisenstein series: I think that at least in a congruence setting, the analytic continuation is in some sense equivalent to analytic continuation of automorphic L function. The Langlands-Shahidi http://en.wikipedia.org/wiki/Langlands-Shahidi_method method deduces from the analytic continuation of Eisenstein series the analytic continuation of automorphic $L$ functions. So every new proof of analytic continuation for Eisenstein series yields a new proof for the analytic continuation of automorphic $L$ functions.