I'd like to prove that the $k$-th suspension of the $n$-disk, i.e $\mathbb{D}^n = \left\lbrace x \in \mathbb{R}^n : \| x \| \leq 1\right\rbrace$ is the $n+k$ disk, i.e
$$S^k\mathbb{D}^n\simeq \mathbb{D}^{n+k+1}$$
I'd like to use Smash product of compact spaces that I was able to prove in order to do the base case(if this is the easier way), but this lead me to :
$$S\mathbb{D}^n \simeq \mathbb{S}^1 \wedge \mathbb{D}^n \simeq (\mathbb{S}^1\setminus \left\lbrace p\right\rbrace \times \mathbb{D}^n\setminus \left\lbrace x_0\right\rbrace)^*$$
But I'm not sure how to proceed from here noticing that this is homeomorphic to $\mathbb{D}^{n+1}$.
From here the thesis quickly follows since $S^k\mathbb{D}^{n} = SS^{k-1}\mathbb{D}^n \simeq S\mathbb{D}^{k-1+n} \simeq \mathbb{D}^{k+n}$ using idnuction hypothesis and the $k=1$ case.
Any help hint or solution would be appreciated.
