Given a sequence of random variables $X_1,X_2,...$ defined on the same probability space $(\Omega,\mathcal{B},P)$.
part 1: Verify that $P(\limsup_{n\to\infty}X_n>x)=0$ if and only if $\lim_{n\to\infty}P(\bigcup_{m=n}^\infty\{X_m>x+\frac1k\})=0$ for all $k\geq 1$
part 2: Suppose $E(X_n)\leq x$ for all $n$ and $\sum_{n=1}^{\infty}\var(X_n)<\infty$.
Prove that $P(\limsup_{n\to\infty}X_n\leq x)=1$
I am guessing that these proofs require the use of Boole and/or Chebychev, but I have always struggled with these.
Help is appreciated!