How to choose which one is correct?

Question

Suppose that $\tau_1$ and $\tau_2$ are topologies on $X$ induced by the metrics $d_1$ and $d_2$, respectively, such that $\tau_1\subseteq \tau_2$. Then which of the following statement is true?

1. If a sequence converges in $(X, d_2)$ then it also converges in $(X, d_1)$.

2. If a sequence converges in $(X, d_1)$ then it also converges in $(X, d_2)$.

3. Every open ball in $(X, d_1)$ is open in $(X, d_2)$.

My try: Since $\tau_1\subseteq \tau_2$, means that every set open in $\tau_1$ is also open in $\tau_2$. Using this it is easy to show that options 2 and 3 are correct.

But the correct answer is option 1. How to choose it?

Answer 1

As for why 1 is true, let $S=(a_n)_{n\in\mathbb{N}}$ be a sequence in $X$ that converges w.r.t $d_2$. Then if $U\in\tau_1$ is open containing the (unique) limit $x$ of $S$, there exists $N\in\mathbb{N}$ with $a_n\in U$ when $n>N$ (because $\tau_1\subseteq\tau_2$). This shows that the sequence converges to $x$ w.r.t $d_1$.