double sequence question: convergence of the average over one index, when the sequence converges over another index

Question

Let $\{a_{N,i}:i\geq 1, N\geq 1\}$ be a real valued double sequence, such that, $$ \lim_{N\rightarrow\infty} a_{N,i} = a_i, \quad\text{for each $i\geq 1$,} $$ where, $a_i\in \mathbb{R}$, for all $i\geq 1$. Can we conclude that, $$ \left|\frac{1}{N}\sum_{i=1}^N a_{N,i} - \frac{1}{N}\sum_{i=1}^N a_i\right| \rightarrow 0,\quad\text{as $N\rightarrow\infty$.} $$ Is there any counterexample to this statement?