Prove using the definition of convergence in probability that $W_n$ converges to $0$ in probability Suppose $W_1, W_2, ...$ is a sequence of random variables.  $W_n$ is defined as the following $W_n = 1/n$ with probability 1/2 and 0 with probability 1/2.  Using the definition of convergence in probability, show that $W_n$ converges to 0 in probability.  So far, I have tried using Chebyshev's and Markov's inequality, but have gotten nowhere.  Any help is appreciated and thanks in advance.

Wikipedia has the definition

A sequence ${X_n}$ of random variables converges in probability towards $X$ if for all $\varepsilon \gt 0$, $\lim_{n\to\infty}\Pr\big(|X_n-X| \geq \varepsilon\big) = 0.$

so you might look at what happens when $1/n \lt \varepsilon$, i.e. when $n \gt 1/\varepsilon$.
