Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two

Statement
Suppose $$M$$ is a connected manifold of dimension at least two. Suppose $$p$$ is a point in $$M$$. Note that the manifold $$M \setminus \{ p \}$$ is still a connected manifold (because $$M$$ has dimension at least two). Consider the inclusion map:

$$i: M \setminus \{ p \} \to M$$

This induces a homomorphism between the fact about::fundamental groups (note that it is not necessary to specify basepoints because both manifolds are path-connected):

$$\pi_1(i): \pi_1(M \setminus \{ p \}) \to \pi_1(M)$$

This induced map is an isomorphism. In particular, both the fundamental groups are isomorphic groups.

Facts used

 * 1) uses::Seifert-van Kampen theorem