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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