In just a couple of years, low-carbohydrate diets have accomplished what the government has failed to do in decades of trying: convince the public that refined grains are bad and whole grains are good.
I don’t know what your ‘progressive tense’ thing is about. Occasionally one hears talk of a ‘present progressive’ in English using an auxiliary (usually to be, never to have, and sometimes a verb of motion) with a present participle, like
- I am walking.
- You’re always complaining.
- She goes sharing her germs with everyone.
- I was walking down the street when the alarm sounded. (past progressive)
- Don’t come running to me. (yes, this is in the imperative)
- I am going to quit. (English’s periphrastic future tense)
That isn’t what we have here. There is no present participle, only a gerund — although those are admittedly identical in form, in that they end in -ing.
The difference is that a present participle “can’t” be the head noun in a noun phrase, whereas a gerund can. Furthermore, English can freely interchange infinitives and gerunds, as both these can be the heads of noun phrases. Consider the equivalence of these two statements, where the first uses gerunds and the second, infinitives:
- Seeing is believing.
- To see is to believe.
That’s all that you see happening here; in the quoted sentence, convince is actually an infinitive, even though it isn’t written with the to particle used in citation forms like to convince. Infinitives can still be infinitives in English without it. Plus that way it forms a parallel construct with the immediately-previous to do; one might even claim that the to particle is distributive in this case. One might even argue that this is one of those slightly unusual cases of (non-finite) verbs in apposition.