Let $X_{n}\sim Bin(n,p_{n})$ and suppose $np_{n}\xrightarrow{n\rightarrow\infty} \infty$. Show that $P(X_n\geq 1)\xrightarrow{n\rightarrow\infty} 1$.

Question

This is a 2-part question.

My answer for part (a). Note that $E(X_n)=np_n$ and that $X_n$ is a nonnegative random variable (since it's binomial). We can use Markov's inequality which tells us $$0\leq P(|X_n|>\epsilon)\leq \frac{np_{n}}{\epsilon^{2}}.$$ Then we let $n \rightarrow \infty$ and use the Sandwich theorem to complete the proof.

Can someone confirm this is right? Also for part (b) I'm struggling to get started, but imagine I have to use Chebyshev's inequality. Is this just normal convergence rather than convergence in probability? I'd appreciate any help.