Resnick Exercise 6.21 - Find expectation and variance of Brownian Motion

Question

Let B be standard Brownian Motion. Define: $$Y_n = \sum_{i=1}^{2n} |B(i/2^n) - B((i-1)/2^n)| $$ How do we show that: $$EY_n = 2^{n/2}E(|B(1)|)$$ $$VarY_n = Var(|B(1)|)$$ $$\sum_n P(Y_n < n) < \infty$$ I'm stuck at the moment, I tried to use the stationary increment property to derive expectation and variance but I am getting a different result, so would love any explanations thanks