N-sphere is (n-1)-connected

Statement

Suppose $n$ is a natural number (i.e., $n\geq 1$). Then, the $n$-Sphere (?) is $(n-1)$-connected. In other words:

• For $n=1$, the $n$-sphere, better known as the circle, is a path-connected space.
• For $n\geq 2$, the $n$-sphere is a Path-connected space (?) and Simply connected space (?) and all its homotopy groups up to the $(n-1)^{th}$ homotopy group are trivial. In other words, $\pi _{1}(S^{n}),\pi _{2}(S^{n}),\dots ,\pi _{n-1}(S^{n})$ are all trivial.

By the Hurewicz theorem, this is equivalent (for $n\geq 2$) to the assertion that $S^{n}$ is simply connected and the first $n-1$ homology groups are trivial.