There is an elementary proof of Peano's existence theorem. Walter, W. Amer. Math. Monthly, 78:170–173, 1971.
bibtex   
@article{MR0276524,
	Author = {Walter, Wolfgang},
	Date-Added = {2010-11-04 10:26:36 -0700},
	Date-Modified = {2010-11-04 10:47:23 -0700},
	Fjournal = {The American Mathematical Monthly},
	Issn = {0002-9890},
	Journal = {Amer. Math. Monthly},
	Keywords = {Peano's existence theorem},
	Mrclass = {34.04},
	Mrnumber = {0276524 (43 \#2268)},
	Mrreviewer = {D. A. Lutz},
	Pages = {170--173},
	Title = {There is an elementary proof of {P}eano's existence theorem},
	Volume = {78},
	Year = {1971},
	Bdsk-Url-1 = {http://www.ams.org/mathscinet-getitem?mr=0276524}}

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