= Solution
Continuity of $\varphi$ at zero and $\varphi(0)=1$ allow $u>0$ so small that
$$
\frac1u\int_{-u}^u(1-\varphi(t))dt<\varepsilon/2.
$$
<Pointwise convergence> and dominated convergence make the corresponding expression for $\varphi_n$ less than $\varepsilon$ for all sufficiently large $n$. Part (ii) then bounds
$$
\mathbb P(|X_n|>2/u)\leq\varepsilon.
$$
There are only finitely many remaining $n$, so enlarge $M\geq2/u$ until each of their tails is also at most $\varepsilon$. This proves uniform tightness.
Back to article page
