Let $\mathbb{S}^m$ and $\mathbb{S}^n$ denote spheres of dimension $m$ and $n$ respectively. Suppose $m<n$ and let $\alpha\in\mathbb{S}^m$ and $\beta \in \mathbb{S}^n$. Assume that $h\colon(\mathbb{S}^m,\alpha)\to(\mathbb{S}^n,\beta)$ is a continuous map that carries $\alpha$ to $\beta$.
I am interested in showing that $h$ is homotopic to the constant map $h_0(x)=\beta$, for all $x\in \mathbb{S}^m$ via maps $h_t$ that carries $\alpha$ to $\beta$. I know that when $m=1$, the result is the same as showing that $\pi_1(\mathbb{S}^n,\beta)$ is trivial for $n>1$. But I am kind of stuck on how to prove this for any $1\leq m<n$.
Any help/hint will be very useful. Thanks.
A well-known result from algebraic topology is that any such $h$ can be deformed to an injective map. See cellular approximation theorem.
Then $h$ (up to homotopy) can't be surjective, otherwise we'd have a homeomorphism (continuous bijection from a compact space to a Hausdorff space).
Then since $h$ misses a point, we have a map into $\Bbb R^n$ ($S^n\setminus \{\text {point}\}\cong\Bbb R^n$). Since $\Bbb R^n$ is contractible, the result follows.