The convergence in the Central Limit Theorem is weak convergence, which is weaker than convergence in probability. I set it as an exercise to find an example that convergence in distribution does not imply convergence in probability:
Let $(X_j)_{j\geq 1}$ be i.i.d. with $E[X_1]=0$ and $\sigma_{X_1}^2=\sigma^2<\infty$. Let $S_n=\sum_{i=1}^nX_i$. Then $ \frac{S_n}{\sigma\sqrt{n}}\to Z\sim N(0,1) $ which is from the CLT.
Here is my question: does $\frac{S_n}{\sigma\sqrt{n}}$ converge in probability?
I think the point is to give a non-zero lower bound of $ P(\frac{S_n}{\sigma\sqrt{n}}>\epsilon) $ for some $\epsilon>0$. But I'm not sure if this can lead to the conclusion that $\frac{S_n}{\sigma\sqrt{n}}$ does not converge in probability.