Convergence in Probability Maximum of a Sample from a Uniform Distribution:
Suppose $X_1, ...  , X_n$ is a random sample for a $\mathrm{uniform}(0,\theta)$ distribution. Suppose $\theta$ is unknown. An intuitive estimate of $\theta$ is the maximum of a sample. Let $Y_n = \max\{X_1, ... , X_n\}$. Exercise 5.1.4 shows that the cdf of $Y_n$ is 
$F_{Y_n}(t) = 1$ if $t>\theta$, $F_{Y_n}(t) = \frac{t^n}{\theta^n}$ if $0 < t \leq \theta$, and $F_{Y_n}(t) = 0$ if $t\le0$. 
Then the pdf of $Y_n$ is $f_{Y_n}(t) = \frac{nt^{n-1}}{\theta^n}$ if $0 < t \leq \theta$, and $f_{Y_n}(t) = 0$ elsewhere.
Based on its pdf, it is easy to show that $E(Y_n) = (n/(n+1))\theta$. Thus, $Y_n$ is a biased estimator $\theta$... Further, based on the cdf of $Y_n$, it is easily seen that $Y_n$ converges to $\theta$ in probability. 
MY QUESTION:

How do we know that $Y_n$ converges to $\theta$ in probability? Is it because $E(Y_n) \rightarrow \theta$?

Thanks in advance.
 A: The definition of convergence in probability is given below :

Let $\{X_n\}$ be a sequence of random variables on a probability space. then we say that $\{X_n\}$ convergence in probability to $\theta$ if for every $\epsilon >0$,
  $$Pr[|X_n-\theta| \geq \epsilon]\rightarrow0  ~\text{as}~ n\rightarrow \infty$$ and this is equivalent to $$Pr[|X_n-\theta|< \epsilon]\rightarrow 1  ~\text{as}~ n\rightarrow \infty$$

For your problem note that $$\begin{align}P[|Y_{(n)}-\theta|< \epsilon] &=P[\theta-\epsilon<Y_{(n)}<\theta+\epsilon]\\ &= F_{Y_{(n)}}(\theta+\epsilon)- F_{Y_{(n)}}(\theta-\epsilon)\\&= 1-(\frac{\theta-\epsilon}{\theta})^n, 0< \epsilon <\theta \\&=1-0,\epsilon  \geq\theta \\& \rightarrow 1 ~\text{as}~ n \to \infty\end{align}$$
Hence $Y_n$ converges to $\theta$ in probability.
A: Assuming $E \left[ Y_n \right] \rightarrow \theta$ so that $E \left[ \left|
Y_n - \theta \right| \right] \rightarrow 0$, you could use Markov's inequality
to show that for $\varepsilon > 0$
\begin{eqnarray*}
  \Pr \left[ \left| Y_n - \theta \right| \geq \varepsilon \right] & \leqslant & E
  \left[ \left| Y_n - \theta \right| \right]/\varepsilon
\end{eqnarray*}
which goes to zero proving convergence in probability.
