'thereby achieving' is correct and usable in written English.
It is used to express a result that follows from something else. For example, "We took steps to reduce costs, thereby achieving greater financial stability."
The McCain campaign has done it, thereby achieving in an instant one of its key objectives.
This linkage engenders potent antitumor effector T cells but inhibits FoxP3\ T reg cells, thereby achieving long-term protective immunity.
Finally, we propose that at steady supply, cartels of competing strategies automatically yield maximum biomass, thereby achieving a collective optimum.
Its control is done by designing a controller input, thereby achieving desirable dynamical behavior.
The proposed performance based PPS has the capability of defeating adversaries thereby achieving its targets.
I must have butt dialled you!", thereby achieving the amazing double feat of making him think of my butt and forcing him to speak to me.
The two "subgroups" release songs at the same time in their respective countries and languages, and promote them simultaneously, thereby achieving "perfect localization," as Lee calls it.
Blessed with a backstory that directly invokes Vlad the Impaler, Oldman plays Dracula as a love-lorn mortal who declares war on God, thereby achieving damned immortality.
It proved a painless matter to fuse them into one humongous global corporation -- thereby achieving efficiency through the largest possible economies of scale.
Wisely, he followed his wife's advice to make one movie "for the bank" and one for his art, thereby achieving an equilibrium.
It very much looks as though the lexicographer used high language and a fancy archaism to "exonerate" dismay, thereby achieving the comfortable sense of having bogged bog.
