1) Although it may usually be better (in order to concentrate better ) to daven Shemona Esrei with closed eyes, in Aseres Yemei Teshuva it is advisable to daven from a siddur, as this way it is less likely to forget the additions that must be added to the Shemona Esrei (Zachreinu L’Chaim, Mi Komocha, Hamelech HaKadosh, Hamelech Hamishpat, U’Ksov and B’Sefer Chaim). (This is practical advice and not a Halacha).
2) If Zachreinu L’Chaim, Mi Komocha, Hamelech Hamishpat, U’Ksov and/or B’Sefer Chaim were mistakenly skipped, and the next Bracha was already started, there is no need to repeat the Shemona Esrei.
However, if "Hamelech HaKadosh" was forgotten and you signed off the Bracha of Ata Kadosh with "HaKeil HaKadosh" like it is said the rest of the year, one must indeed repeat the Shemona Esrei, as the main point of these Yomim Noraim is to realize that Hashem is our King and to accept His Kingdom (and all the Mitzvos that come along with it) upon ourselves.
Thus forgetting "Hamelech HaKadosh" is forgetting the essence and "missing the point" of these significant days, and therefore must be repeated.
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Our holy sages, the Chazal tell us
תנא דבי אליהו כל השונה הלכות בכל יום מובטח לו שהוא בן עולם הבא, שנאמר הליכות עולם לו, אל תקרי הליכות אלא הלכות - one who studies [at least two] Halachos daily is assured a portion in Olam Haba - the world to come. (:מגילה כ"ח )
Important Numbers
