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How can we know that a black hole exists if it attracts both matter and light, and to prove that it exists would need to observe it?

It's a very common question but I don't know the answer.

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So far, astronomers relies on indirect observations to detect black holes. Black holes' enormous gravitational pull not only attracts matter into its event horizon, but also strongly influences matter close to the event horizon.

We may have chances to detect black holes' Hawking radiation directly during the last stage of the evaporation of light (primordial) black holes (created in early universe, much less massive than other black holes), but these attempts all failed. Check the wikipedia article about some details

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