Nazir 1:7 states:
הֲרֵינִי נָזִיר כְּמִנְיַן יְמוֹת הַחַמָּה, מוֹנֶה נְזִירוּת כְּמִנְיַן יְמוֹת הַחַמָּה. אָמַר רַבִּי יְהוּדָה, מַעֲשֶׂה הָיָה, כֵּיוָן שֶׁהִשְׁלִים מֵת:
[If one says], “Behold I am a nazirite, as the number of days in a solar year” he must count as many naziriteships as there are days in the solar year. Rabbi Judah said: such a case once occurred, and when the man had completed [his naziriteships], he died.
R' Ovadia MiBartenura explains that R' Yehuda is telling this story to prove R' Yehuda Hanasi wrong about one of his opinions in a previous Mishna, Nazir 1:4.
הֲרֵינִי נָזִיר כִּשְׂעַר רֹאשִׁי, וְכַעֲפַר הָאָרֶץ, וּכְחוֹל הַיָּם, הֲרֵי זֶה נְזִיר עוֹלָם וּמְגַלֵּחַ אַחַת לִשְׁלשִׁים יוֹם. רַבִּי אוֹמֵר, אֵין זֶה מְגַלֵּחַ אַחַת לִשְׁלשִׁים יוֹם. וְאֵיזֶהוּ שֶׁמְּגַלֵּחַ אַחַת לִשְׁלשִׁים יוֹם, הָאוֹמֵר הֲרֵי עָלַי נְזִירוּת כִּשְׂעַר רֹאשִׁי, וְכַעֲפַר הָאָרֶץ, וּכְחוֹל הַיָּם:
[If one says,] “Behold, I am a nazirite as the hairs of my head”, or “As the dust of the earth”, or “As the sands of the sea,” he becomes a life-nazirite, and shaves his head every thirty days. Rabbi (R' Yehuda Hanasi) says: this one does not shave his head every thirty days. The one who shave his head every thirty days is the one who says, “Behold, upon me are naziriteships as the hair on my head”, or “As the dust of the earth”, or “As the sands of the sea.”
R' Yehuda Hanasi is saying that when one says that he will be a nazir forever, it is one long naziriteship, and he will only be allowed to shave once every twelve months. The Tanna Kama says that when one says that he will be a nazir forever, it is an infinite number of consecutive standard naziriteships. Because a standard naziriteship lasts for thirty days, the nazir may shave once a month.
In Nazir 1:7, the Tanna Kama says that one who says he will be a nazir like the days of the year will have consecutive standard naziriteships equal to the number of days in a solar year, or 365, and can shave once a month. R' Yehuda Hanasi would argue that it is one long naziriteship, and therefore he will be a nazir for life, and can shave once a year. R' Yehuda disproves this by telling of a man who had made such a promise, spent 365 months as a nazir, completed his naziriteship, and died.
The Mishna doesn't waste words. The nazir completing his naziriteship is enough to disprove R' Yehuda Hanasi, who would say that the nazir never completes his naziriteship. Why does R' Yehuda need to mention that the nazir died?