I am trying to prove that $\Bbb{R}^n$ minus finitely many points $x_1,\ldots,x_m$ is simply connected, where $n \geq 3$. For days now I have tried many different arguments but I have found flaws in all of them. I have finally come up with one, except that there is some small detail that I need to know how to prove.
I prove that $\Bbb{R}^n$ minus finitely many points is simply connected by inducting on the number of points that I remove. If I remove one point (the case $m=1$), I get that $\Bbb{R}^n - \{x_1\} \cong S^{n-1} \times \Bbb{R}$ which upon applying $\pi_1$ shows me that $\Bbb{R}^n - \{x_1\}$ is simply connected.
Inductive Hypothesis: Now suppose that $\Bbb{R}^n -\{x_1,\ldots,x_k\}$ is simply connected for all $ k
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Suppose now I have $m$ points $x_1,\ldots,x_m$ lying in $\Bbb{R}^n$. Now if I write each of these points out in coordinates, I know that there is at least one $j$ with $1 \leq j \leq n$ such that the $j-th$ coordinate of all my points are not all the same (otherwise my points are all just one point and there is nothing to prove!).
Because I have only finitely many points, suppose without loss of generality that $x_1$ is the point whose $j-th$ coordinate is the greatest (this does not mean that such a choice is unique, I only want to know if it exists). Say that the $j-th$ coordinate of $x_1$ is $c$. Now I consider the plane
$\mathbf{x}_j = \{\mathbf{x} \in \Bbb{R}^n : \text{$j$ -th coordinate of $\mathbf{x}$ is equal to $c$} \}.$
My idea now is to apply the Seifert-Van Kampen Theorem together with the induction hypothesis as follows: I set
$A = \Bigg\{\mathbf{x} \in \Bbb{R}^n- \{x_1,\ldots,x_m\} : \text{$j$ -th coordinate of $\mathbf{x}$ is greater than $c-\varepsilon$} \Bigg\}$
where $\varepsilon$ is chosen such that $A$ does not enclose all my points and
$B = \Bigg\{\mathbf{x} \in \Bbb{R}^n- \{x_1,\ldots,x_m\} : \text{$j$ -th coordinate of $\mathbf{x}$ is less than $c $} \Bigg\}.$
Here's a picture of what I'm trying to do in the case of $\Bbb{R}^3$: \Bbb{R}^3">
Then $A$ is open and so is $B$, clearly $A$ and $B$ are path connected and their intersection which is just a "cuboid" being a convex set is path connected as well.
My Problem: I want to apply my inductive hypothesis to $A$ and $B$ in order to deduce that $\pi_1(A) = \pi_1(B) = 0$. The problem now is that the inductive hypothesis is for $\Bbb{R}^n$ and not "chopped off bits" of $\Bbb{R}^n$ like $A$ and $B$. How do I get around this? Can I say that $A$ and $B$ are somehow deformation retracts of $\Bbb{R}^n$ minus finitely many points?
Thanks.