Let $X_{1}$, $X_{2}$,...$X_{n}$ be independent random variables that are uniformly distributed over [-1,1]. Show that $$ Y_{n}= X_{n}/n $$ Converges to some limit, and identify the limit.
I have
$$ P(|Y_{n}|\geq \varepsilon)=P(|X_{n}/n|\geq \varepsilon) <P(|1/n|\geq \varepsilon)\rightarrow 0 $$
When I look at the solution, I saw that the proof is concluded with:
"...for all $n$ with $ \dfrac1n < \varepsilon$."
It appears there is contradiction in my proof because $\varepsilon$ can not be both less or equal to $1/n$ and greater than $1/n$.
What have I done wrong?
Assuming that you are trying to show convergence in probability, we have for any $\varepsilon > 0$ $$P(|Y_{n}|\geq \varepsilon)=P(|X_{n}/n|\geq \varepsilon) = P(|X_n| \geq n\varepsilon) = 0 \; \; \forall \; n \ge \frac{1}{\varepsilon}$$
Therefore, $\lim_{n\to\infty} P(|Y_{n}|\geq \varepsilon) \to 0$ for any $\varepsilon > 0$.