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 Fundamental group (?)s (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. Seifert-van Kampen theorem