Let $\bar{Y}_n$ denote the sample average of $n$ samples from a distrubution with mean $\mu$ and variance $\sigma^2$. Assume sampling was done indepedently. Consider an estimator of $\mu$, $W_n = \frac{n-1}{n}\bar{Y}_n$. Find the Probability limit of $W_n$.
My attempt: Intuitively, $\bar{Y}_n$ converges almost surely (a.s.) to $\mu$, using the Strong Law of Large numbers and $\lim_{n \to \infty} \frac{n - 1}{n} = 1$ so $W_n$ should converge to $\mu$. But I couldn't work out a proof that uses the fromal $\epsilon$-defintion of Convergence in Probability. Another idea was to use the Central Limit theorem then use the CDF of normal distribution to work out a closed form for $\epsilon$. I'm stuck mainly because I'm not able to "fit" the $\frac{n - 1}{n}$ into the requirements "a sequence of iid RV's with finite $\mu$ and $\sigma^2$" of any of the theorems. Any help writing the formal proof will be greatly appreciated.