The phrase "assumed that eventually" is correct and usable in written English.
It can be used when discussing a belief or expectation that something will happen after a period of time.
Example: "She assumed that eventually, the project would be completed despite the delays."
Alternatives: "expected that in due time" or "believed that ultimately".
I know Cherry was, at one point, booking a lot of car washes in and around the Bay Area, and I always assumed that, eventually, the purpose of raising venture capital was to expand their labor force and upsell other car-related services to become a more diversified business.
"When we started Priceline, we assumed that eventually the Internet would foster sites like that," Mr. Ek said of BiddingForTravel.
People assumed that eventually there would be peace and only took very few things with them as they thought they'd be able to go back.
They may well have assumed that eventually the FBI would learn the truth and drop the investigation, as the Bureau in fact did -- but in the meantime they were harassing Jim, and, one suspects, hoping to intimidate him.
For each of the 18 countries we assumed that eventually the rate of improvement of mortality rates will converge towards a common rate of decline.
I assumed that would eventually occur in any event".
But let's assume that, eventually, Mose will work as intended.
But the government and Kadin assume that eventually they will develop the ability to handle the fallout and that in any case, Indonesia gets a bigger share of a smaller pie – and it's their pie.
It will continue to make set-top boxes that bring in programs from cable and the Internet, but it assumes that eventually all televisions will simply connect directly to home networks.
Everyone assumes that eventually hundreds of billions of dollars will be made off Internet technology, which is why America Online could acquire Time Warner in the first place, but no one knows exactly how this windfall will be reaped.
