Are there two-sided $\varepsilon$-expanders with independent sets of size $(1-\varepsilon)n$?

Terry Tao's notes on expander graphs has the following exercise:

Exercise 13 Let $G$ be a $k$-regular graph on $n$ vertices that is a two-sided $\epsilon$-expander for some $n > k \geq 1$ and $\epsilon>0$. Show that any independent set in $G$ has cardinality at most $(1-\epsilon) n$.

Is this bound known to be tight?