Being a novice to this topic, I am not being able to properly write down a step by step solution to this problem.
For each integer $n$, let $X_n$ be a non negative random variable with finite mean $\mu _n$ . Prove that if $\underset{n \rightarrow > \infty}{\lim}{\mu _n}=0 $ then $X_n \mbox{ converges in probability to }0$.
Update: Please don't down vote if my solution is wrong. Of course you are welcome to edit and correct it...
This is a solution I tried following @whuber’s instructions. I don’t know whether I have gone wrong somewhere or not.
Assume $$ X_n \space doesnot\space converge\space in \space probability \space to\space 0$$ $$\therefore\space there\space exists\space \epsilon \gt 0\space such \space that$$ $$\lim_{n\to\infty} P\left[|X_n-\mu_n|\ge\epsilon\right]\rightarrow 1$$ $$\implies \lim_{n\to\infty} P\left[X_n \ge\mu_n +\epsilon\right]\rightarrow 1$$ $$\implies \lim_{n \to\infty} \frac{\left(E\left[X_n\right]=\mu_n\right)}{\mu_n + \epsilon}\rightarrow 1$$ $ \implies\lim_{n\to\infty} \mu_n\to\mu_n + \epsilon$ which is a contradiction as $\epsilon$ is a number greater than 0. Hence our initial assumption was wrong. Hence proved.