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