prove that X converges in probability to 1

Question

How to prove that $X_n \xrightarrow[]{p} 1$:

$X_n = 1 + nY_n$, where $Y_n$ is a Bernoulli random variable with mean $1/n$

Answer 1

Hint: $$ P(|X_n - 1|> \epsilon) = P(nY_n > \epsilon) = P(Y_n > \epsilon/n) = 1-P(Y_n\le \epsilon/n). $$