= Solution
A subspace $Y\subseteq X$ is a <retract> of $X$ if there is a continuous retraction
$$
r:X\to Y
$$
such that $r(y)=y$ for every $y\in Y$. A space is a <contractible space> if its identity map is <homotopy>[homotopic] to a constant map.
Let $i:Y\hookrightarrow X$ be the inclusion and let
$$
H:X\times I\to X
$$
be a contraction, with $H(x,0)=x$ and $H(x,1)=x_0$. Then
$$
K(y,t)=r(H(i(y),t))
$$
is continuous and satisfies
$$
K(y,0)=r(i(y))=y,
\qquad
K(y,1)=r(x_0).
$$
\b[Thus $K$ contracts $Y$, proving that every <retract of a contractible space> is contractible.]
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