How to prove a probability inequality in Feller's book? Let $X_1, X_2$ be independent real-valued random variables with their tail distributions satifying $1-F_i(x)\sim x^{-\rho}L_i(x)$, where $\rho>0$ and $L_i(x), i=1,2$ are slowly varying functions. Let $t'=(1+\delta)t>0, \delta>0, t>0$. For every $\varepsilon>0$ and $t$ large enough,
$$P(X_1+X_2>t)\geqslant (1-\varepsilon)[P(X_1>t')+P(X_2>t')].$$
How to prove this inequality?
Remark 1. Let $L(x)$ be a positive monotone function on $[0, \infty)$ to $[0, \infty)$. We say $L(x)$ be slowly varying at $+\infty$ if $L(tx)/L(x)\to 1$ as $x\to \infty$ for every $t>0$.
Remark 2. This inequality comes from W. Feller. An Introduction to Probability Theory and
Its Applications. Wiley & Sons, 1971, page 279, (8.15).
 A: After two days of thinking, I finally came up with the answer to this question.
Firstly, we must understand the following properties:
Property 1.  If $P(C_n)\to 1$, then for every $\varepsilon>0$ and $n$ large enough,
$$P(D\cup C_n)\geqslant 1-\varepsilon\geqslant P(D)(1-\varepsilon).$$
Property 2. (i) $A\cup(B\cap C)=(A\cup B)\cap (A\cup C)$.
(ii) $A\cap(B\cup C)=(A\cap B)\cup (A\cap C)$.
Property 3.  Since $\{X_1>t', X_2>-\delta t\}\subset \{X_1+X_2>t\}$ and
$\{X_2>t', X_1>-\delta t\}\subset \{X_1+X_2>t\}$, so
$$\{X_1+X_2>t\}\supset \{X_1>t', X_2>-\delta t\}\cup \{X_2>t', X_1>-\delta t\}.$$
Secondly, we start to prove the inequality we wanted.
Proof. Define
$$A=\{X_1>t'\}, B=\{ X_2>-\delta t\},$$
$$ C=\{X_2>t'\}, D=\{ X_1>-\delta t\}.$$
Note that
$$A\cup D=D, B\cup C=B, B\cup D=\{X_1>-\delta t\}\cup\{X_2>-\delta t\},$$
and
$$(A\cap B)\cup (C\cap D)=(A\cup C)\cap (A\cup D)\cap(B\cup C)\cap (B\cup D).$$
It's easy to see that, as $t\to \infty$,
$$P(A\cup D)=P(D)\to 1, P(B\cup C)=P(B)\to 1, P(B\cup D)\to 1.$$
From the Property 3, 2 and 1 we can get that
$$P(X_1+X_2>t)\geqslant P(A\cap C)(1-\varepsilon).$$
Now the inequality we wanted follows from the independence of $X_1$ and $X_2$, that is
$$P(X_1+X_2>t)\geqslant [P(X_1>t')+P(X_2>t')](1-\varepsilon).$$
