Cfrgtkky

I'm trying to prove SLLN(4):
Let ${X_n : ngeq1}$ be a sequence of $L^1-$integrable independent random variables on a probability space and $S_n = sum_{j=1}^nX_{j}$ for every $ngeq1$. Let $phi : mathbb{R} to mathbb{R}$ be a positive and continuous even function such that $frac{phi(x)}{x}$ is non-decreasing in $x in (0,infty)$ and $frac{phi(x)}{x^2}$ is non-increasing in $x in (0,infty)$. Assume that for some sequence ${b_n: ngeq1}$ of positive real numbers with $b_n to infty$ as $n to infty$,
$$sum_{ngeq1}frac{mathbb{E}[phi(X_n)]}{phi(b_n)} <infty$$
Prove that
$$frac{S_n-mathbb{E}[S_n]}{b_n} to 0 qquad mbox{a.s.}$$

In order to finalize the proof I just need to show
$$frac{T_n-mathbb{E}[T_n]}{b_n} to 0 quad mbox{a.s.}$$
where $T_n = sum_{j=1}^{n}Y_j$, and $Y_j$ is truncated version of $X_j$:
$$Y_n = X_n mathbb{1}_{|X_n|le b_n},$$ which in turn, only suffices to show $sum_{n=1}^{infty} frac{operatorname{Var}(Y_n)}{b_n^2} < infty$.

How can I prove this last inequality?

For a non negative random variable $X$, the following inequalities hold:
$$
X^2mathbf 1left{Xleqslant bright}leqslant bXmathbf 1left{Xleqslant bright}
=bfrac{X}{phileft(Xright)}phileft(Xright)mathbf 1left{Xleqslant bright}
leqslant bfrac{b}{phileft(bright)}phileft(Xright),$$
where we used the fact that $xmapsto phi(x)/x$ is non-decreasing.
Apply this to $X=leftlvert X_nrightrvert$ and $b=b_n$ to get the wanted result.