On page 58-59 of the notes by Knill (found here :http://www.math.harvard.edu/~knill/teaching/math144_1994/probability.pdf ) there is a version of the WLLN whose proof I have trouble understand.
On page 59, I don't understand how we go from $$ ||S_n||_1 \le ||S_n^{(R)}||_1+ ||T_n^{(R)}||_1 $$
to
$$\le ||S_n^{(R)}||_2 + 2 \sup_{1\le l \le n } E[|X_l| : |X_l| \ge R] $$
I know I am supposed to use the Minkowski and Cauchy-Schwarz inequalities but I can't work out the details. Please show me how.
The bound $\lVert S_n^{(R)}\rVert_1 \leqslant \lVert S_n^{(R)}\rVert_2$ is indeed a consequence of Cauchy-Schwarz inequality.
For the second term, notice that $$T_n^{(R)}=\frac 1n\sum_{i=1}^n X_i\mathbf 1\{|X_i|\gt R\}-\mathbb E\left[X_i\mathbf 1\{|X_i|\gt R\}\right],$$ hence taking the $L^1$-norm, $$\lVert T_n^{(R)}\rVert_1\leqslant \frac 2n\sum_{i=1}^n\mathbb E\left|X_i\mathbf 1\{|X_i|\gt R\}\right|.$$ Since for each $i\in\{1,\dots,n\}$, we have $$\mathbb E\left|X_i\mathbf 1\{|X_i|\gt R\}\right|\leqslant \max_{1\leqslant l\leqslant n}\mathbb E\left[|X_l|\mathbf 1\{|X_l|\gt R\} \right],$$ we get the wanted inequality.