I'm trying to understand this exercise:
Let $X_1,X_2,...$ be a sequence of random variables such that $\max_{1 \leq k \leq n} \{|X_k|\} \to_{\mathbb{P}} 0$. Show that this sequence satisfies the weak law of large numbers.
If I understood this law correctly, then I need to prove that $\frac{X_1+...+X_n}{n} \to_{\mathbb{P}} \rm{E}(X_n)$. but my teacher solved this problem proving that $\rm{P}(|\frac{X_1+...+X_n}{n}| > \epsilon) \to 0$. How does this solve the problem? Wasn't it supposed to be proved that $\rm{P}(|\frac{X_1+...+X_n}{n} - \rm{E}(X_n)| > \epsilon) \to 0$
Your teacher's conclusion is right if you replace the condition with "$\max_{k\geq n}|X_k|\overset{P}{\to}0$". We pick $\delta>0$ and $\epsilon>0$ arbitrarily first.
"$\max_{k\geq n}|X_k|\overset{P}{\to}0$" implies that $$P(\bigcup_{k\geq n}\{|X_k|\geq\delta\})\leq P(\{\max_{k\geq n}|X_k|\geq\delta\})\to0,\ n\to\infty.$$
Choose a $N$ such that $P(\bigcup_{k\geq N}\{|X_k|\geq\delta\})\leq\epsilon/2$.
\begin{eqnarray} P\left(|\frac{1}{n}\sum_{k=1}^{n}X_k|\geq\delta\right)&\leq& P\left(|\frac{1}{n}\sum_{k=1}^{N-1}X_k|\geq\delta\right)+P\left(|\frac{1}{n}\sum_{k=N}^{n}X_k|\geq\delta\right)\\ &\leq&P\left(|\frac{1}{n}\sum_{k=1}^{N-1}X_k|\geq\delta\right)+P\left(\bigcup_{k\geq N}\{|X_k|\geq\delta\}\right)\\ &\leq& \epsilon\quad\text{(when $n$ is big enough)}. \end{eqnarray}
The key to understand this is that $\omega\in\Omega$ where $X_k(k\geq n)$ are relatively large will overall squeeze in a small set when $n\to\infty$.