Let $X_t$ be a sequence of i.i.d. RVs with mean $0$ and unit variance. Then $S_n=n^{-1/2}\sum_{t=1}^n X_t$ converges in distribution to a $N(0,1)$. We know that $S_n$ does not converge in probability: the typical argument is to compare $S_{2n}$ to $S_n$ and show they are not getting closer in probability as $n$ increases. However, if I compare $S_{n+1}$ to $S_n$ I get
$$S_{n+1}-S_n=\left(\frac{\sqrt{n}}{\sqrt{n+1}}-1\right)S_n+\frac{X_{n+1}}{\sqrt{n+1}}=o(1)O_p(1)+o_p(1)=o_p(1)$$
My question is why this does not imply that $S_n$ converges in distribution? Am I using the Cauchy criterion wrongly?