2

Is there a single word that represents a 'growing conviction' or 'increasingly convinced'?

For example:

Montag's growing conviction of the importance of books is evident by the end of the first section.

or

Montag becomes increasingly convinced of the importance of books by the end of the first section.

2

persuade [per-sweyd] verb (used with object), per·suad·ed, per·suad·ing.

  1. to induce to believe by appealing to reason or understanding; convince:

Source: Dictionary.com

“Montag is persuaded about the importance of books by the end of the first section.”

1

Montag's certainty of the importance of books is evident by the end of the first section.

certainty TFD

  1. The fact, quality, or state of being certain, especially: a. Inevitability: the certainty of death. b. The quality of being established as true: the certainty that the earth orbits the sun. c. Confidence; assurance: his certainty that things would get better.
0

You can use "acceptance" or a synonym of it.

acceptance:

  1. Agreement with or belief in an idea or explanation. Oxford

So from your sentences:

Montag's acceptance of the importance of books is evident by the end of the first section

0

'Burgeoning' is a good word for what you are trying to convey. However, you would still need 'conviction': 'burgeoning conviction'. 'Burgeoning' means something budding - beginning to grow. https://en.oxforddictionaries.com/definition/burgeon

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