TITLE: Pointwise convergence of $X_n$ vs $X_nI_{\{|X_n|\leq c_n\}}$ and of $\sum X_n$ vs $\sum X_nI_{\{|X_n|\leq c_n\}}$
QUESTION [3 upvotes]: Let $\{X_n,n\geq 1\}$ be a sequence of random variables, and $\{c_n,n\geq1\}$ a positive sequence. Let also $\sum_n P(|X_n|> c_n)<\infty$. Prove:

If $Y_n=X_nI_{\{|X_n|\leq c_n\}}$ and $P\left( {\mathop {\lim }\limits_n {Y_n} = X} \right) = 1$, then $$P\left( {\mathop {\lim }\limits_n {X_n}} =X\right) = 1$$
$$P\left( {\sum\limits_n {{X_n}} } \,\text{converges}\,\right) = P\left( {\sum\limits_n {{X_n}{I_{\left\{ {\left| {{X_n}} \right| \le {c_n}} \right\}}}} } \,\text{converges}\,\right)$$


I have tried to prove it by the following method. 
First, by Borel-Cantelli Theorem, we have $$P\left( {\mathop {\lim \sup }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) = 0.$$
And$$0 \le P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) \le P\left( {\mathop {\lim \sup }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) = 0,$$then $$P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) = 0.$$
Since $P\left( {\mathop {\lim }\limits_n {Y_n} = X} \right) = 1$,we obtain $$P\left( {\mathop {\lim }\limits_n \left| {{X_n}} \right| \le {c_n}} \right) = 1.$$ Hence
$$P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu}  {{X_n}} } \right) = P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| > {c_n}} \right) + P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} \left| {{X_n}} \right| \le {c_n}} \right) = 1.$$
Then $$1 = P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu} {X_n}} \right) \le P\left( {\mathop {\lim }\limits_n {\mkern 1mu} {X_n} = X} \right) \le 1,$$
which means $$P\left( {\mathop {\lim }\limits_n {X_n}} =X\right) = 1.$$
It seems something wrong! Thank everyone for good ideas!

REPLY [1 votes]: By Hypothesis, $\sum_n P(\{X_n\neq Y_n\})<\infty$. 
By Borel-Cantelli Lemma, $P\left( {\mathop {\lim \inf }\limits_n {\mkern 1mu}  {{\{X_n=Y_n\}}}} \right) = 1.$ 
Put $A={\mathop {\lim \inf }\limits_n {\mkern 1mu}  {{\{X_n=Y_n\}}}}$
For every $\omega \in A, X_n(\omega)=Y_n(\omega)$ for all large n.
This shows, for every $\omega \in A,$  $$\mathop {\lim}\limits_n X_n(\omega)=\mathop {\lim}\limits_n Y_n(\omega)$$
$$\sum_n X_n(\omega)\;converges \Leftrightarrow \sum_n Y_n(\omega)\;converges$$ 
Now, the required results follow easily.
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