Convergence in probability 
Can anyone tell me how they got the regions $0<\epsilon<\theta$ and $\epsilon >0 $.
Also to clarify, is the last step where it says $\lim_{n \to \infty} P(|Y_n-\theta|>\epsilon)=0$

Imagine we have $X$ representing the toss of a fair die.
Then $\text{Range}(X) = \{1,2,\cdots,6\}$
$$P(X = x) = \frac16$$
So:
$$P(X \le 1) = \frac16$$
$$P(X \le 2) = \frac26$$
$$P(X \le 3) = \frac36$$
$$P(X \le 4) = \frac46$$
$$P(X \le 5) = \frac56$$
$$P(X \le 6) = \frac66 = 1$$
Also:
$$P(X \le 7) = \frac66 = 1$$
$$P(X \le 0) = \frac06$$
$$P(X \le 0.5) = \frac06$$
$$P(X \le -2000) = \frac06$$
$$P(X \le 4.5) = \frac46$$
$$P(X > 7) = \frac06$$
Thus:
The '$\varepsilon > \theta$' (I'll assume '$\varepsilon > 0$' is a typo) is like '$X > 7$'
The '$0 < \varepsilon < \theta$' (I'll assume '$\varepsilon > 0$' is a typo) is like '$1 < X < 6$'

Also
Show that $Y_n$ converges in probability
