A vanishing-probability spike — Borel–Cantelli settles a.s. convergence

Suppose $\{X_n,n\ge1\}$ is a sequence of random variables such that $P[X_n=0]=1-\dfrac1{n^3}$ and $P[X_n=n]=\dfrac1{n^3}, n\ge1$. Which of the following is not true?