You mean "Books help US..."
Yes, both examples are correct. I would certainly say either "deeper" or "more deeply," though I think that from a strictly grammatical standpoint "deeper" should be in the adverb form.
November 1, 2018
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Books help us to think harder and more deeply.
You need an adverb for think which harder fulfills but deep needs to be changed to deeply and needs more to add a superlative.
November 1, 2018
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