"becoming bored" is correct and can be used in written English.
For example, "After learning the same material for weeks on end, I was becoming bored with the class."
By the time of the O.E.D.'s first citation of the noun "boredom" in 1852, in Dickens's "Bleak House" (where it occurs six times by my count), everyone, or at least everyone in the novel-reading middle classes, seemed to be bored, or worried about becoming bored.
America was becoming bored with their antics.
He says the album is, in part, the result of becoming bored with his own self-obsession.
frozen sweets division in the United States, but said he quit a year later to join a rival, M pick, after becoming "bored with ice cream".
and watching the accused parties squirm — and before the fourth stage of everybody becoming bored because nothing ever changes in a system rotten to the core.
Nixon, his ambitions undimmed despite the California loss, also offered a way out of a law career of which Garment was becoming bored.
Because my daughters are grown, I used to worry about my friend's younger children becoming bored when they stopped by to visit.
If the game remained this easy, however, we'd quickly tire of the jellybeans and gum drops, becoming bored after a couple of binge sessions.
COLUMBUS, Ohio — Senator Barack Obama was an hour and a half late to a rally last week at Ohio State University, and the crowd was becoming bored.
Export of Lebkuchen has remained fairly stable over the same period, indicating the decline is mainly due to Germans becoming bored of traditional sweets.
He hasn't had to worry, though, about becoming bored at Fila USA, a unit of Fila Holding S.p.A., the Italian sportswear company.
