Exercise 1.2.19 of Hatcher's Algebraic Topology I've been trying to prove exercise $1.2.19$ of Hatcher's algebraic topology:

Show that the subspace of $\mathbb{R}^3$ that is the union of the spheres of radius $\frac{1}{n}$ and center $(\frac{1}{n}, 0, 0)$ for $n = 1, 2, ···$ is simply-connected.

I was thinking of applying Seifert-Van Kampen inductively somehow, but I can't conclude using just that argument. I saw this proof but I would like to see a proof of this exercise without using $1-$skeletons.
How can I prove this using just Seifert-Van Kampen?

Let $S_n$ be the sphere of radius $1/n$, and let $U_n$ be an open neighborhood of the origin $(0,0,0)$ within $S_n$. Then we can cover the space $X$ by $V_n=(\bigcup_{n=1}^\infty U_n)\cup S_n$ for $n=1,2,\cdots$. Since $V_n$ are path-connected and have trivial fundamental groups, the intersection of any two or more $V_n$'s is $\bigcup_{n=1}^\infty U_n$, which is also path-connected and has trivial fundamental group, $\pi_1(X)=*_n\pi_1(V_n)=0$.
