I'm stuck with this problem.
Let $X_1, X_2, ...$ be a sequence of independent Bernoulli random variables. Show that if $\sum_{i=1}^n \frac{p_i}{n} \to l \; \text{ as } \; n \to \infty$ then $\sum_{i=1}^n \frac{X_i}{n} \to l \; \text{ stochastically }$
I have noticed some facts, if we call $X'_n = \sum_{i=1}^n \frac{X_i}{n}$, then $\mu_n = E[X'_n] = \sum_{i=1}^n \frac{p_i}{n}$, so in some sense the $\mu_n$ converges to $l$.
Also using the usual method of proving the Bernoulli law of large numbers I have arrived at the following $P[|X'_n - \mu_n| < \epsilon] \geq 1 - \frac{\sum_{i=1}^np_iq_i}{\epsilon^2n^2}$ And the rightmost term I think it goes to $0$ as $n \to \infty$, so this would almost give me the result except for the $\mu_n$ which would have to be replaced by $l$ (which seems reasonable given that the $\mu_n$ converges to it). I'm a little rough in this area so I'm unsure of which manipulations I can do, and also I am unaware of a lot of theorems (maybe one can lead to the result I want).
So anyway, my question is mainly, I am going in the right direction, and if so, how can I conclude the result?
Or are the other standard ways (like using moment generating functions or calculating the cumulative distribution of the $X'_n$ and taking limits) better approaches to this problem?
Any help would be appreciated :)