This is how I approached this:

Let $\displaystyle S = \sum_{j=1}^{\infty} d_{2j} + d_{2j+1}, \ T = \sum_{j=1}^{\infty} d_{2j+1} + d_{2j+2}$ and $\displaystyle W_n = \sum_{j=1}^{n} d_j$.

Then the partial sums $\displaystyle S_n = \sum_{j=1}^{n} d_{2j} + d_{2j+1}$ and $\displaystyle T_n = \sum_{j=1}^{n} d_{2j+1} + d_{2j+2}$ converge.

Observe that $\displaystyle \forall n \geq 1, \ S_n = W_{2n+1} - d_1 \ \text{and } \forall n \geq 2, \ T_{n-1} = W_{2n} -d_1 -d_2$. Then from the convergence of $\displaystyle S_n , \ T_n$ we get that $\displaystyle W_{2n} \text{ and } W_{2n+1}$ converge (You can actually prove it with only having that $\displaystyle W_{2n}$ converges and $\displaystyle d_n \to 0$..)

Now you finish it