Properties of a sequence of iid rv's 
I cannot do part a), and Im fairly sure that b) and c) will follow from it. If possible could I please have a solution to part a) and hints if you feel necessary to parts b) and c).
 A: Write $$n \, 1_{(X_1^2>\epsilon^2n)}\leq {X_1^2\over\epsilon^2} \, 1_{(X_1^2>\epsilon^2n)}.$$ Now integrate both sides, and use 
dominated convergence to conclude  $n\mathbb{P}(|X_1|>\epsilon \sqrt{n})\to0.$

The sequence of random variables ${X_1^2\over\epsilon^2} \, 1_{(X_1^2>\epsilon^2n)}$
 converges to zero pointwise as $n\to\infty$. Since they are dominated by the integrable random variable $X_1^2$, the dominated convergence theorem guarantees that 
$$\mathbb{E}\left({X_1^2\over\epsilon^2} \, 1_{(X_1^2>\epsilon^2n)}\right)\to0 \mbox{ as } n\to\infty. $$
A: For (a): consider $Y$ a simple random variable, that is,a linear combination of characteristic functions of measurable sets. Then 
$$n\mathbb P\{|X_1|\gt \varepsilon\sqrt n\}\leqslant n\mathbb P\{|X_1-Y|\gt \varepsilon\sqrt n\}+n\mathbb P\{|Y|\gt \varepsilon\sqrt n\},$$
hence using Markov's inequality, 
$$n\mathbb P\{|X_1|\gt \varepsilon\sqrt n\}\leqslant\frac 1{\varepsilon^2}\mathbb E[(X_1-Y)^2]+n\mathbb P\{|Y|\gt \varepsilon\sqrt n\}.$$
Since $Y$ is bounded, the second term of the RHS is $0$ for $n$ large enough, hence
$$\limsup_{n\to +\infty}n\mathbb P\{|X_1|\gt \varepsilon\sqrt n\}\leqslant\frac 1{\varepsilon^2}\mathbb E[(X_1-Y)^2].$$
