Strong law of large numbers and the central limit theorem

The strong law of large numbers and the central limit theorem use different modes of convergence. Is it nevertheless true that the strong law of large numbers can be shown from the central limit theorem?

It seems that, if random variables $$\sum_i X_i$$ converge to the gaussian distribution in probability, then due to the fact we normalized them for this to happen, that if we did not normalize them, then they would converge almost surely to a dirac delta. But the details elude me.

For example, suppose $$X_n$$ are iid standard normal random variables, and $$U$$ uniform on $$[0,1]$$ independent of the $$X_n$$. Let $$T_n = n$$ if $$U$$ is in an interval $$[a_n, a_n+1/n] \subset [0,1]$$ where the $$a_n$$ are arranged so every point of $$[0,1]$$ is in infinitely many of the intervals. Let $$Y_n = X_n + T_n - T_{n-1}$$ (with $$Y_1 = X_1 + T_1$$). Then the partial sums $$S_n = \sum_{i=1}^n Y_n = T_n + \sum_{i=1}^n X_n$$ satisfy the conclusion of the Central Limit Theorem in that $$S_n/\sqrt{n}$$ converges in distribution to the standard normal distribution, but does not satisfy the conclusion of the Strong Law of Large Numbers in that $$S_n/n$$ does not converge a.s. to $$0$$, indeed almost surely $$\limsup_{n \to \infty} S_n/n \ge 1$$.