What is the Uniform Convergence Problem in Stochastic Functions?

[St41n]

First of all, hello all this is my first post i think. Congratulations on this great community
Please move my post if I'm not posting on the right forum and I'm sorry for any inconvenience.

I have this problem that I need to solve and I don't have a clue. I hope you could give me some ideas.

I need to show this: $$\mathop {\sup }\limits_\theta \left\| {E_\theta \left( {\hat \beta _T } \right) - b\left( \theta \right)} \right\| \to 0$$

knowing that:
$$\hat \beta _T \stackrel{T\rightarrow\infty}{\rightarrow} b\left( {\theta _0 } \right)$$

where $$\hat \beta _T$$ is a stochastic function of $$y_T$$ that comes from a distribution with true parameter $$\theta_0$$

θ and β belongs in a compact subset of R^p and R^q respectively.

The convergence apparently is non-stochastic as we've taken expectation.
A hint is to add and subtract something into the norm and use the triangle inequality to show the above claim. But, I have no idea how to treat the expectation.

I haven't supplied all info there is, but please tell me if you can think of any possible approaches for this. Any help is much appreciated

 

[Focus]

I get the feeling you should add and subtract $$\hat \beta_T$$ and use its convergence and the strong law of large numbers maybe.