The induced metric

[keen geode]

Hello,
I can't understand what makes the proof that long. (Pic 1: The largest)
Isn't that sufficient to prove that The open sets of (𝐴,𝑑_𝐴) are exactly the intersections of the open sets of (𝑋,𝑑) with 𝐴? (Pic 2: The smallest)
Since the open balls are considered a kind of basis for the whole topology.
Can anyone clarify, please?

[lament void]

Well pic 2 doesn't even claim anything about open sets

And yes, the fact it's true for open balls implies it's true for open sets because balls form a basis is good intuition, but rigorously showing that is exactly what the rest of pic 1 is