Multiplication and inversion preserve the identity component in a Lie group

Question

Let $G_0$ be the connected component of the identity in a Lie group $G$, and let $\mu$ and $i$ be the multiplication map and the inverse map of $G$. I would like to show that for any $x \in G_0$

1. $\mu(\{x_0\} \times G_0) \subset G_0$
2. $i(G_0) \subset G_0$.

For (1), since multiplication is smooth in a Lie group it is necessarily continuous. Moreover, $\{x_0\} \times G_0$ is connected since it is the product of a singleton set and a connected set. Hence, the image of $\mu$ must be connected. I am stuck on showing that this connected image must be a subset of $G_0$. For the second problem, I assume we can use the same reasoning since $i$ is also assumed to be smooth in a Lie group.

Answer 1

Hint You haven't yet used the hypotheses that $1 \in G_0$ and $x \in G_0$. Notice that $$\mu(\{x\} \times G_0) \ni \mu(x, 1) = x .$$