The proof I was given in class for Weierstrass Theorem using Berstein Polynomials contains this one particular inequality:
$|f(x) − E[f(\frac{S_n}{n})]| ≤ E[|f(x) − f(\frac{S_n}{n})|]$
E stands for expected value, x is the probability of success of a sequence of n iid Bernoulli variables, and Sn is the sum of those n iid Bernoulli variables.
I understood the theorem and how to prove it using Weak Law of Large Numbers, but I'm facing trouble in understanding the inequality I stated above.
Please help!
$f(x)$ is a fixed number. It is not random. So $f(x)-E(\frac {S_n} n)=E[f(x)-E(\frac {S_n} n)]$. Now use the inequality $|EY| \leq E|Y|$.