Definition. Define $\eta\in C^\infty (\mathbb{R}^n)$ by
$$\eta(x) := \begin{cases} C\exp\left(\frac{1}{\lvert x \rvert^2-1}\right), & \mbox{if } \lvert x \rvert< 1 \\ 0 & \mbox{if } \lvert x \rvert\ge 1\end{cases}$$
the constant $C>0$ selected so that $\int_{\mathbb{R}^n}\eta\;dx =1$.
For each $\varepsilon > 0$, set $$\eta_\varepsilon (x):= \frac{1}{\varepsilon^n}\eta\left(\frac{x}{\varepsilon}\right).$$ The functions $\eta_\varepsilon$ are $C^\infty$. Evans book Partial Differential Equation said that $$\text{supp}(\eta_\varepsilon)\subseteq B(0,\varepsilon).\tag1$$
My Solution
Suppose that $x\in\text{supp}(\eta_\varepsilon)$, then $\eta_\varepsilon (x)\ne 0$ and therefore $$ \left| \frac{x}{\varepsilon} \right|<1$$ and the $x\in B(0,\varepsilon)$.
Question 1. It's correct?
Now we observe that $\eta_\varepsilon (x)\ne 0$ for all $|x|<\varepsilon$, then by closed support definition we have that $$\text{supp}(\eta_\varepsilon)=\overline{B}(0,\varepsilon).\tag2$$
Question 2. Aren't $(1)$ and $(2)$ contradictory? Where am I wrong? What definition of support does Evans use at this point?