Learn English – “I want you succeeding” vs. “I want your succeeding”

I am happy seeing you succeed.

This means, effectively, I am happy while I'm seeing you succeed. There is a slight implication of causality -- I am happy because I see you succeeding -- but not that strong.

For example, if I said, "I am happy waiting here", I would mean that waiting is acceptable, that it would not make me unhappy to wait while the other person went and did something.

I am happy to see you succeed.

This could mean exactly what you think -- I am happy because I see you succeeding -- but in other constructions it could mean I am volunteering for something, or saying I would stand by while it happened. "I am happy to help you move" is me volunteering to help. "I am happy to watch you kill yourself" is a way of saying I won't interfere.

I want you to succeed.

This can only be interpreted literally.

I want you succeeding.

This means "I want to maintain you in the state of being successful". A doctor might say, "I want you exercising and eating right" -- he wants you to do that every day.

I want your succeeding.

This is grammatically correct but not idiomatic. "Your succeeding" is a legitimate noun-phrase, but saying "I want it" usually means I want to possess something, not that I want that thing to occur. So "I want a birthday party" sounds childish; says, "I want to have a birthday party."

"Your succeeding is important to me" is more idiomatic, but using the gerund "succeeding" as a noun sounds odd when there is an independent noun "success" that could be used instead.