-Every observation we have analysed has a material cause.
-By inductive reasoning, all observations have materialistic causes.
-Hence, there are no immaterial causes and immaterialism doesn't exist.
(Unobservable immaterialism are unfalsifiable (because they are unobservable) and hence not considered)
Is this a sound argument?
Immaterial causes are unfalsifiable, so your argument doesn't/can't falsify them. It just tries to convince, but cannot prove anything.
The first part is also questionable. There are plenty of things where we don't know the cause, can't observe it, or there is none (radioactivity is spontaneous - while you can argue it has a material cause, there is nothing that triggers decay, it just happens following some statistics). A lot of things we can't tell if what we see is causation or correlation.
I don't want to judge your statements based on logic (seems weak here from strict math logic), I just want to provide my perspective for your benefit.
Your claim is actually most educated people think, kind of like materialism and empiricism, rejecting the existence of any non-material spirits or Platonic Forms. Well, same large amount of people of most religions will offer you the opposite claim, some may go to extreme and claim "there's nothing but God".
My own perspective is that there's actually no conflict between materialism/empiricism vs idealism. Their only difference is how they define the word "exists"! If two equal saints meet and argue about something, most likely u should bet they're both honest, smart and wise guy, just confused about each other's definition. This universal confusion caused so many unnecessary conflicts throughout history.
The famous "Problem of Universals" in philosophy circle is such an ancient typical conflict, now I think every sensible person will agree there exists a concept called "2" after seeing two seemingly identical things. The only critical insight in this simple-yet-seemingly-philosophically-deep problem is that all human concepts/definitions are relative, meaning the concept "exists" itself is relative to a certain layer of our mind... "2" is certainly more abstract than "pain" which our body can feel impressively while a number, not so much due to our biological design. While certain math genius may feel "2" more impressively than "pain", and that's why this person can surely outperforms you in solving Math number theory problems...
By the way, most people will regard math as absolute truth, such as number 2 is real and really exists in some Platonic spiritual world apart from this imperfect material world (sounds like dualism here). But my view is contrary, number 2 (or any abstract universal concepts) resides in the same "metaphoric" realm of human mind, just happen to be located in the relatively most clear-countably-verifiable-universal layer. I don't opine separating the noumena from the phenomena as a serious business, its useful for some purposes, but all these concepts and separations are still man-made (fake) analogies and stories consciously engineered to explain to a naive but confused child who is actively seeking an authoritative answer from the grown-ups.
This world perceived by human mind is nothing but metaphors, that's why we can have several different models/theories about the same phenomena, such as the famous Newtonian Force Laws, Lagrangian/Hamiltonian Minimum Action Principle, and the later Maxwell/Einstein Field Theory in classical physics and then applied further into QM, so far all these above 3 distinct models (metaphors) are not proved wrong and taught in every physics department around the globe. In the meantime, because our mind is constantly forming-destroying-reforming numerous metaphors as free will, most of these created images/processes/analogies are in more or less confused state. For example, if you've never been visiting a place and people around you are talking extensively about it, still in your mind you'll form some vague images from what you heard. Most of these misconceptions are like "avidya" in Buddhism metaphoric teachings, huge huge and thick darkness in the form of ignorant confusions is covering human mind like five mountains and thus all its derived senses...
