3 verbs next to each other

3 verbs next to each other
 

I'm doing a homework that is about Perífrasis. It's verbs + gerundio such as:
Andar, acabar, estar, ir, llevar, seguir, venir + gerundio and the different signficances.

The sentence is as shown below.
Necesito _______________________ (seguir, convencerla) de que no hay peligro alguno porque.......

I wrote down:
Necesito seguir convénciendola (not sure where accent goes) de que no hay...

My roommate, who does not speak any English said it should be
Necesito seguirle convénciendola de que no hay...

Not sure why LE is needed and why it would go after the 2nd verb and not be added to the gerundio or before the 1st verb. Also, I thought that when there are 3 verbs together, it should be broke up by a 'de' between the 2nd and 3rd verb if the 3rd wasn't a gerundio. In this case, it is so I would think it would be between the 1st and 2nd. I never actually learned a rule to dealing with 3 verbs together but this is what I assume/observed. I was once told that you can just avoid having 3 verbs by rewording the sentence but this is a fill in the blank question.
http://i.imgur.com/mCTNY1d.jpg

Well, first off, you don't have three verbs together. The conjugated word is the verb, followed by an infinitive and a gerundio (English: present participle).

Except for the placement of the accent mark, what you wrote is correct. You should not lose the stressed syllable in the gerundio when suffixing a pronoun.
(The accent mark goes over the same vowel you marked in sentence 2 of your homework.)

That "le" in "seguir" isn't necessary and it sounds really bad. The point four "(estar, ser)" [está siendo] is something that I would never say, but "se está convirtiendo" or "se está volviendo".

A pleasure.

I agree that "seguirle convenciéndola" is not correct.
This is very colloquial speech and grammatically incorrect.

Thanks everyone. Yeah, I thought it was weird too. I read it as "is being" which sounds very weird in that context. I was thinking "is becoming/changing" sounds better. I don't fully understand the whole idea of volver in this context but a classmate said to me last week that the significance is "to become" (something in the future). It seems to fit well here.

Yes. "Volver" means "to come back" and its pronominal form (volverse) means "to become".

Good luck!