Oliver Straser is correct in that Milnor himself in 1956 [1] only showed his model for $EG$ is weakly contractible (that is, all its homotopy groups vanish) with his coarse topology on the join. (For any compact Hausdorff spaces $X$ and $Y$, by the tube lemma, note that Milnor's $X \circ Y$ equals the quotient-topology join $X \ast Y$.)

Historically, for any topological group $G$, the first proof in the literature of the contractibility of Milnor's $EG := G^{\circ \aleph_0}$ is contained within the two-page proof of Theorem 8.1 in Dold's 1963 article [2].

A few years later in 1966, tomDieck in Section 7 of [3] offers a more conceptual argument.

Contemporarily, I agree that the comment of Oldřich Spáčil is an acceptable answer, based on the simplified proof that tomDieck later gives in Proposition 14.4.6 of his 2008 book [4], which rehashes his earlier works.

[1] : John Milnor, *Construction of universal bundles II*, Annals Math 63(3):430–436, 1956.

[2] : Albrecht Dold, *Partitions of unity in the theory of fibrations*, Annals Math 78(2):223–225, 1963.

[3] : Tammo tomDieck, *Klassifikation numerierbarer Bündel*, Arch Math (Basel) 17:395–399, 1966.

[4] : Tammo tomDieck, *Algebraic Topology*, 2008.

contractible? $\endgroup$ – johndoe Dec 4 '13 at 8:41T. tom Dieck: Algebraic Topologythe space $E_{G}$ is indeed contractible and the assumption on $G$ is only that it is a topological group (as far as I can understand). The argument is pretty much the same as the one mentioned in the above comment by few_reps. $\endgroup$ – Oldřich Spáčil Dec 4 '13 at 15:33