Proving convergence in probability of a rescaled random variable. 
Let $\lambda_n=1/n$ for $n=1,2,\ldots$.Let $X_n \sim \text{Poisson}(\lambda_n)$.
(a) Show that $X_n \xrightarrow{\text{P}} 0$.
(b)Let $Y_n =nX_n$. Show that $Y_n \xrightarrow{\text{P}} 0$.

For part (a) I've used Markov's inequality
$$\mathbb{P}(\left | X_n \right |>\varepsilon )\leq \frac{\mathbb{E}[X_n]}{\varepsilon }=\frac{\lambda _n}{\varepsilon }=
\frac{1}{n\varepsilon }\xrightarrow[n\rightarrow \infty ]{ } 0$$
But for part (b) nothing seems to work.
Markov's inequality leads to
$$\mathbb{P}(\left | Y_n \right |>\varepsilon )\leq \frac{\mathbb{E}[nX_n]}{\varepsilon }=\frac{1}{\varepsilon }
$$
Neither does Chernoff's method
$$ \mathbb{P}\left (X_n>\frac{\varepsilon}{n}  \right ) =\mathbb{P}\left (e^{tX_n}>e^{t\varepsilon/n}  \right ) \leq \frac{\mathbb{E}[e^{tX_n}]}{e^{t\varepsilon/n}}=  \frac{e^{(e^t-1)/n}}{e^{t\varepsilon/n}}= \xrightarrow[n\rightarrow \infty ]{ } 1 $$
And if I haven't done any mistakes, $Y_n$ does not converge in $L^p$ to $0$.
At this point I'm stuck. I'd appreciate any advice.

You can do it using only elementary facts about $(X_n)_n$: fixing any $\varepsilon >0$, for $n\geq 1$
$$\begin{align}
\mathbb{P}\{|Y_n| > \varepsilon\}
&=\mathbb{P}\{Y_n > \varepsilon\}\\ 
&\leq \mathbb{P}\{Y_n > 0\} \\
&= \mathbb{P}\{X_n > 1/n\} \\
&= \mathbb{P}\{X_n \geq 1\}  \tag{$X_n$ is integer-valued}\\
&= 1-\mathbb{P}\{X_n =0\}\\
&= 1-e^{-\lambda_n}\\
&= 1-e^{-1/n}\\
&\xrightarrow[n\to\infty]{} 0
\end{align}$$
On the other hand, you are right, I don't see how to use the MGF approach to easily make this work.
