Probability question that links between geometric and exponential distribution with limits.

Let $\lambda >0$, and for every $n\ge 1$, we have ${X}_{n}\sim \mathrm{Geo}(\frac{\lambda}{n})$.

We define ${\hat{X}}_{n}=\frac{1}{n}{X}_{n}$. Show that for every $t\ge 0$:

${F}_{{\hat{X}}_{n}}(t)\to {F}_{\hat{X}}(t)$ when $n\to \mathrm{\infty}.$. while $\hat{X}\sim \mathrm{Exp}(\lambda )$.

Let $\lambda >0$, and for every $n\ge 1$, we have ${X}_{n}\sim \mathrm{Geo}(\frac{\lambda}{n})$.

We define ${\hat{X}}_{n}=\frac{1}{n}{X}_{n}$. Show that for every $t\ge 0$:

${F}_{{\hat{X}}_{n}}(t)\to {F}_{\hat{X}}(t)$ when $n\to \mathrm{\infty}.$. while $\hat{X}\sim \mathrm{Exp}(\lambda )$.