Can you solve $y'(x) = y^2(x)\;\land\;y(x_0) = y_0 $ on $]-1, 1[$ using the Banach contraction theorem?

Question

I would like to solve the following Cauchy problem:

$$ \begin{cases} y'(x) = y^2(x)\\ y(x_0) = y_0 \end{cases}\tag 1 $$ In my opinion, using the Banach contraction theorem it can only be solved in $I:= \ ]-1, 1[$ since:

$$\forall x_1, x_2 \in \mathbb R: |x_1^2 - x_2^2 | < |x_1 -x_2| \Longrightarrow x_1, x_2 \in I \tag 2$$

In that puprose, $x_0$ must also be in $I$. Is my reasonment correct ?