In Lee's Introduction to Topological Manifolds, problem 7-12 asks to show that $\{(1,0)\}$ is a deformation retract, but not a strong deformation retract of the subspace of the plane
$$ X = \bigcup_{n=0}^\infty L_n$$
where $L_0$ is the line segment connecting $(1,0)$ to the origin and $L_n$ is the line segment connecting the origin to $(1,1/n)$ for $n \geq 1$. The part I'm struggling with is to show that it is not a strong deformation retract. If it were, then there would be a continuous map $H: X \times [0,1] \rightarrow X$ with
$$H(x,0) = x, \quad \forall x \in X;\\ H(x,1) = (1,0), \quad \forall x \in X; \\ H((1,0),t) = (1,0) \quad \forall t \in [0,1]$$
I was hoping a contradiction would pop out somewhere, but no luck. Can someone give a hint or suggest a different approach?