Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables where \(X_{n}\) has distribution \(F_{n}\) which has mean \(\theta_{n}\) for all \(n \in \mathbb{N}\). Suppose that

\[\lim _{n ightarrow \infty} \theta_{n}=\theta,\]

for some \(\theta \in(0, \infty)\). Hence it follows that

\[\lim _{n ightarrow \infty} E\left(X_{n}ight)=E(X),\]

where \(X\) has distribution \(F\) with mean \(\theta\). Does it necessarily follow that the sequence \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is uniformly integrable?