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とうとう,
遂(つい)に is an adverb that is frequently used to mean ‘finally’, or ‘at last’. It will usually come at the beginning of a sentence.
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遂(つい)に日本(にほん)上陸(じょうりく)!!!
Finally it has arrived in Japan!!!
遂(つい)にドラゴンズが勝(か)った!
The Dragons have finally won!
遂(つい)に may be seen equally as often with or without kanji. When compared to 到頭(とうとう), ついに is slightly more formal.
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遂(つい)に最後(さいご)の日(ひ)ですね、今(いま)まで色々(いろいろ)とありがとうございました。
Finally it is your last day. Thank you for everything up until now.
とうとう今日(きょう)で最後(さいご)だね。今(いま)まで色々(いろいろ)とありがとう。
It's finally your last day. Thank you for everything up until now.
The kanji 遂(つい) has a meaning similar to ‘to get (something) over with’. Due to this, ついに will most often be used to highlight things that have finally happened after a significant period of time, or a long course of events. It will not be used to express things that do not require time or effort.
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遂(つい)に結婚(けっこん)したのか!おめでとう! (Natural)
You finally married!? Congrats!
昨日(きのう)貰(もら)った袋(ふくろ)を遂(つい)に捨(す)てた。 (Unnatural)
I finally threw away the bag I bought yesterday.