Let $\{X_i\}_{i=1}^n$ be a sequence of uncorrelated random variables, $$ \text{Cov}(X_i,X_j)=0, $$ with $$ E(X_i)=\mu_i<\infty, $$ and $$ \frac{V(X_i)}{i} \xrightarrow{i\to \infty} 0. \tag{1} $$ Show $$ \frac{ \sum_{i=1}^n \left(X_i -\mu_i\right)}{n} \xrightarrow{L^2} 0. $$
Using the martingale
$$
M_n = \sum_{i=1}^n \frac{X_i-\mu_i}{i},
$$
which is uniformly integral if
$$
\sum_{i=1}^{\infty} \frac{V(X_i)}{i} < \infty. \tag{2}
$$
Then, by the martingale convergence theorem, $M_n$ converges a.s. to some finite random variable $M$.
By Kronecker lemma
$$
\sum_{i=1}^n \frac{i}{n} \frac{\left(X_i -\mu_i\right)}{i} \xrightarrow{a.s.} 0
$$
Problem is, condition (2) is very different than (1), and I get almost sure convergence instead of $L^2$ convergence.
Squaring $\sum_{i=1}^n (X_i-\mu_i)$ and integrating, we get $\sum_{i=1}^n V(X_i)$ because covariances vanish. So, the goal is to show $\sum_{i=1}^n V(X_i)=o(n^2)$, given that $V(X_i)=o(i)$. This is pretty standard: given $\epsilon>0$, find $k$ such that $V(X_i)<\epsilon \, i$ for $i>k$. Then $\sum_{i=1}^k V(X_i)=O(1)$ and $\sum_{i=k+1}^n V(X_i)<\epsilon \sum_{i=k+1}^n i< \epsilon\, n^2$