Verifying convergence in probability and distribution. Suppose random variables $\{X_n\}$ are defined on $([0,1],\mathcal{B}([0,1]),\lambda)$ (where $\lambda$ is Lebesgue measure) as follows:
$X_1 = \mathbb{1}_{[0,1]}, X_2 = \mathbb{1}_{[0,1/2]}, X_3 = \mathbb{1}_{[1/2,1]}, X_4 = \mathbb{1}_{[0,1/3]}, X_5 = \mathbb{1}_{[1/3,2/3]}, X_6 = \mathbb{1}_{[2/3,1]}$, etc.
Exercise:
a) Verify that $X_n \stackrel{p}{\to} 0$
b) Verify that for any $w \in [0,1]$, $X_n \not\to 0.$
Questions:


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*a) How do I show that $\lim\limits_{n \to \infty}P(\left|X_n\right|>\epsilon)\to 0$? Intuitively it feels very clear that $X_n$ converges in probability: as $n$ goes to infinity the probability that the indicator functions is equal to $1$ goes to zero. I'm not sure however how to mathematically show this.

*b) What does $w$ do for our random variables? $X_n$ isn't dependent on $w$ right? So how can we write $X_n(w)$?  

a) Note that 
$P(x_n=1)=\frac{1}{k}$ for all $\frac{(k-1)k}{2}<n\leq \frac{k(k+1)}{2}$.
Hence as $n\to\infty$ (because $k\to\infty$) 
$P(|X_n|>\epsilon)\to 0$.
b) Here note that 
for any $w$,
$X_n(w)=1$ for infinitely many $n\in\mathbb N$.
Hence $X_n(w)\not\to0$.
