Let be $X_1,X_2$ two i.i.d binomially distributed random variables, where $p$ is the probability of success and $m$ the length of the underlying Bernoulli experiment. In a proof our professor argues $$ \lim\limits_{n\to\infty}P\left(\left|\frac{X_1+X_2}{n}\right|\geq \epsilon\right)=0, $$ because of Chebyshev's inequality.
I don't understand why she needs Chebyshev's inequality and how we could apply it here? We know that $X_1,X_2$ are both bounded by $m$, so \begin{align*} P\left(\left|\frac{X_1+X_2}{n}\right|\geq \epsilon\right)\leq P\left(\left|\frac{2m}{n}\right|\geq \epsilon\right). \end{align*} Hence, for all $n\geq \frac{m}{\epsilon}$ the inequality $\left|\frac{2m}{n}\right|\geq \epsilon$ is never true so that $P\left(\left|\frac{2m}{n}\right|\geq \epsilon\right)=0$. This means that $\lim\limits_{n\to\infty}P\left(\left|\frac{2m}{n}\right|\geq \epsilon\right)=0$.
Am I missing something?