How to directly show convergence in probability

Question

$\hat{\theta_n}$ is an estimator for $\theta$.

Let $\hat{\theta_n}$ be $\theta$ with probability $\frac{n-1}{n}$ and let it be $n k$ with probability $\frac{1}{n}$. ($k$ is a constant.)

How can we directly prove or disprove $\hat{\theta_n} \to_p \theta$?

By directly I mean to show $\lim_{n \to \infty} P(|\hat{\theta_n} - \theta| \geq 0) = 0$ holds or not.

Thank you!

Answer 1

$$\hat{\theta}_n - \theta = \begin{cases} 0 & W.P. \frac{n-1}{n} \\ nk-\theta & W.P. \frac{1}{n}\end{cases}$$

Hence $\lim_{n \to \infty} P(|\hat{\theta_n} - \theta| \geq 0) \leq \lim_{n \to \infty} \frac{1}{n}$

Answer 2

$$ \lim _{n\to \infty}P(|\hat{\theta}_n-\theta|>\epsilon)= \lim _{n\to \infty}P(\hat{\theta}_n=nk) = \lim _{n\to \infty}\frac{1}{n} = 0. $$