sequences and series - Convergence/divergence of the limit of a sum

In order to prove asymptotic normality of the time series OLS estimator (this context is not important), I have done the following:

Assumptions:

• $\mathrm{var}\left(y_{t}\right)=\gamma_{0}\quad\forall t$

• $\mathrm{cov}\left(y_{t},y_{t-j}\right)=\gamma_{j}\in\mathbb{R}^{k}\quad\forall j$

• $\underset{j\rightarrow\infty}{\lim}\mathrm{cov}\left(y_{t},y_{t-j}\right)=\gamma_{j}=0$

• ${\sum}\left|\gamma_{j}\right|<\infty$

Then:

$$\begin{align} L&= \underset{{\scriptstyle T\rightarrow\infty}}{\lim}\mathrm{\mathrm{var}}\left(\sqrt{T}\left(T^{-1}\sum_{t}y_{t}-\mu\right)\right) \\ &= \underset{{\scriptstyle T\rightarrow\infty}}{\lim}\mathrm{\mathrm{var}}\left(T^{-1/2}\sum_{t}y_{t}\right) \\ &=\underset{{\scriptstyle T\rightarrow\infty}}{\lim}T^{-1}\mathrm{\mathrm{var}}\left(\sum_{t}y_{t}\right) \\ &=\underset{{\scriptstyle T\rightarrow\infty}}{\lim}T^{-1}\sum_{t}\sum_{s}\mathrm{cov}\left(y_{t},y_{s}\right) \\ &=\underset{{\scriptstyle T\rightarrow\infty}}{\lim}T^{-1}\left[T\cdot\gamma_{0}+2\cdot\left(T-1\right)\cdot\gamma_{1}+2\cdot\left(T-2\right)\cdot\gamma_{2}+...+2\cdot\gamma_{T-1}\right] \\ &=\underset{{\scriptstyle T\rightarrow\infty}}{\lim}T^{-1}\left[T\gamma_{0}+2\sum_{1\leq j\leq T-1}\left(T-j\right)\gamma_{j}\right] \\ &=\underset{{\scriptstyle T\rightarrow\infty}}{\lim}\gamma_{0}+2\sum_{1\leq j\leq T-1}\left(1-\dfrac{j}{T}\right)\gamma_{j} \end{align}$$ I'm assuming the last step equals: $\gamma_{0}+2\sum_{j=1}^{\infty}\gamma_{j}$

since concluding from here is rather easy. The problem being that I do not think it's the case, since j also converges to infinity.

Am I wrong? If so, why?