Let $X \sim \text{Ber}(1/2)$. I am asked to show that the sequence of random variables $\{X_n = (1+\frac{1}{n})X\}$ converges in probability to $X$. My attempt:
Let $\epsilon > 0$, take $N = \frac{1}{\epsilon}$, then for all $n > N$ we have $$\begin{cases} & X_n = 0 \text{ wp } 1/2 \\ & X_n = 1+\delta \text{ wp } 1/2 \end{cases}$$ Where $\delta < \epsilon$ for all $n > N$. Hence $P(|X_n-X| > \epsilon) = 0$ and the result follows. Is this enough to prove it? I am familiar with convergence of sequences of functions from real analysis but am having a harder time with random variables. Also, I know that almost surely convergence implies convergence in probability. Can I use that fact here?