If $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ then $\sum_{n=1}^{\infty}(a_{n}-a_{n+2})$ converges
Using the fact that $|a_{n}-a_{m}|\leq \frac{1}{\min{(m, n)}}$ I showed that $a_{n}$ is Cauchy hence $a_{n}\to L$.
If we let $b_{n}=a_{n}-a_{n+2}$ then by looking at the partial sums: $S_{2n}=\sum_{k=1}^{2n}b_{k}=a_{2}-a_{2n}\to a_{2}-L$ and $S_{2n-1}=\sum_{k=1}^{2n-1}b_{k}=a_{1}-a_{2n-1}\to a_{1}-L$
But I seem to missing something because if $a_{1}\neq a_{2}$ the above partial sums don't converge to the same limit and the series diverges. Where is the error?