How is rate of decay proportional to the number off nuclei?

I'm not familiar with the terminology "activity", but according to websites I just checked, "activity" is the same as "decay rate". Here for instance.

http://www.iem-inc.com/prhlfr.html

It's not how many times one nucleus breaks down. A given nucleus only decays once in its life. Then it's a different nucleus. It does so at a random time. One nucleus has a certain probability of decaying in a given time. The more unstable it is, the more likely it is to decay in a given time.

The half-life is the time such that any given nucleus has a 50% chance of decaying in that time. So when you have a bunch of nuclei, pretty close to half of them will decay in that time. If you have 1000 nuclei, 500 of them will decay. If you have 10000 nuclei, 5000 of them will decay. "Half of N" is obviously proportional to N.

Decay rate is closely related, but the unit of time is a fixed interval like a second. Suppose one nucleus has a 1% chance of decaying in a second. Then with 1000 of them, on average 10 will decay in a second. With 10000 of them, 100 will decay in a second. Again, "1% of N" is obviously proportional to N.

Edit: Your terminology of "nuclei in a nucleus" shows you have a little confusion about just what's breaking down. A nucleus is a collection of protons and neutrons bound together. The number of protons is the atomic number. That's what we use to name the nucleus. If there are 92 protons, it's called a Uranium nucleus. When it decays, it breaks down into smaller collections of protons and neutrons. If it sends off a Helium nucleus (2 protons, 2 neutrons), that leaves 90 protons behind and what is left is now called a Thorium nucleus, element 90.

Wording is everything.
activity and rate of decay are closely related.

A nucleus can only break down ONCE.
So your first statement should have read " the FRACTION of the nucleii that breaks down in a given time period"
Almost the same but not quite.

And for the second, the higher the activity the smaller the fraction of nucleii that would be left undecayed after a given amount of time.

Or the higher the activity the greater the fraction of nucleii that HAVE decayed after a given amount of time

What you said is basically correct. But your wording is incorrect. Activity is directly related to decay. You might want to look up beta decay or weak interaction they are the same. It is the action of nucleus  decay. Beta decay is a phenomena and closely related to electromagnetism A single inter action called electro weak interaction. It is a force acting on a scale of the atomic nucleus