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 $(n - 1)^{t h}$ homotopy group are trivial. In other words, $π_{1} (S^{n}), π_{2} (S^{n}), \dots, π_{n - 1} (S^{n})$ are all trivial.

By the (what 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.