Sequence of independent random variables...

The text in the graphic is as follows:

\(\displaystyle \mbox{Let }\, X_1,\, X_2,...\,\mbox{ be a sequence of independent r.v.'s}\)

\(\displaystyle \mbox{with }\, EX_i\, =\, \mu_i,\, Var(X_i)\, =\, {\sigma_i}^2,\, i\, =\, 1,\, 2,....\)

\(\displaystyle \mbox{Assume that }\)

\(\displaystyle \mbox{i) }\, \lim_{n\rightarrow \infty}\, \frac{1}{n^2}\, \sigma_{i=1}^n\, {\sigma_i}^2\, =\, 0\)

\(\displaystyle \mbox{ii) }\, \lim_{n\rightarrow \infty}\, \frac{1}{n}\, \sigma_{i=1}^n\, \mu_i\, =\, \mu\)

\(\displaystyle \mbox{Show that for every }\, \epsilon\, >\, 0,\)

lim⁡n→∞ P(∣σi=1n Xin − μ∣ > ϵ) = 0\displaystyle \lim_{n\rightarrow \infty}\, P\left(\left|\frac{\sigma_{i=1}^n\, X_i}{n}\, -\, \mu\right|\, >\, \epsilon\right)\, =\, 0
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