Editing 0.999... (section)

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Please note that the content of this article is well-established. If you disagree with the underlying mathematics behind this article, or have an argument against one or more of the proofs listed here, please read the FAQ on Talk:0.999..., or discuss it on Talk:0.999.../Arguments, before attempting to edit here.
For mathematicians: please understand that the earlier, more naive proofs are not as rigorous as the later ones as they intend to appeal to intuition, and as such may require further justification to be complete. Thank you.
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Copy and paste: – — ° ′ ″ ≈ ≠ ≤ ≥ ± − × ÷ ← → · §   Cite your sources: <ref></ref>


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Symbols: ~ | ¡ ¿ † ‡ ↔ ↑ ↓ • ¶   # ∞   ‹› «»   ¤ ₳ ฿ ₵ ¢ ₡ ₢ $ ₫ ₯ € ₠ ₣ ƒ ₴ ₭ ₤ ℳ ₥ ₦ ₧ ₰ £ ៛ ₨ ₪ ৳ ₮ ₩ ¥   ♠ ♣ ♥ ♦   𝄫 ♭ ♮ ♯ 𝄪   © ¼ ½ ¾
Latin: A a Á á À à  â Ä ä Ǎ ǎ Ă ă Ā ā à ã Å å Ą ą Æ æ Ǣ ǣ   B b   C c Ć ć Ċ ċ Ĉ ĉ Č č Ç ç   D d Ď ď Đ đ Ḍ ḍ Ð ð   E e É é È è Ė ė Ê ê Ë ë Ě ě Ĕ ĕ Ē ē Ẽ ẽ Ę ę Ẹ ẹ Ɛ ɛ Ǝ ǝ Ə ə   F f   G g Ġ ġ Ĝ ĝ Ğ ğ Ģ ģ   H h Ĥ ĥ Ħ ħ Ḥ ḥ   I i İ ı Í í Ì ì Î î Ï ï Ǐ ǐ Ĭ ĭ Ī ī Ĩ ĩ Į į Ị ị   J j Ĵ ĵ   K k Ķ ķ   L l Ĺ ĺ Ŀ ŀ Ľ ľ Ļ ļ Ł ł Ḷ ḷ Ḹ ḹ   M m Ṃ ṃ   N n Ń ń Ň ň Ñ ñ Ņ ņ Ṇ ṇ Ŋ ŋ   O o Ó ó Ò ò Ô ô Ö ö Ǒ ǒ Ŏ ŏ Ō ō Õ õ Ǫ ǫ Ọ ọ Ő ő Ø ø Œ œ   Ɔ ɔ   P p   Q q   R r Ŕ ŕ Ř ř Ŗ ŗ Ṛ ṛ Ṝ ṝ   S s Ś ś Ŝ ŝ Š š Ş ş Ș ș Ṣ ṣ ß   T t Ť ť Ţ ţ Ț ț Ṭ ṭ Þ þ   U u Ú ú Ù ù Û û Ü ü Ǔ ǔ Ŭ ŭ Ū ū Ũ ũ Ů ů Ų ų Ụ ụ Ű ű Ǘ ǘ Ǜ ǜ Ǚ ǚ Ǖ ǖ   V v   W w Ŵ ŵ   X x   Y y Ý ý Ŷ ŷ Ÿ ÿ Ỹ ỹ Ȳ ȳ   Z z Ź ź Ż ż Ž ž   ß Ð ð Þ þ Ŋ ŋ Ə ə
Greek: Ά ά Έ έ Ή ή Ί ί Ό ό Ύ ύ Ώ ώ   Α α Β β Γ γ Δ δ   Ε ε Ζ ζ Η η Θ θ   Ι ι Κ κ Λ λ Μ μ   Ν ν Ξ ξ Ο ο Π π   Ρ ρ Σ σ ς Τ τ Υ υ   Φ φ Χ χ Ψ ψ Ω ω   {{Polytonic|}}
Cyrillic: А а Б б В в Г г   Ґ ґ Ѓ ѓ Д д Ђ ђ   Е е Ё ё Є є Ж ж   З з Ѕ ѕ И и І і   Ї ї Й й Ј ј К к   Ќ ќ Л л Љ љ М м   Н н Њ њ О о П п   Р р С с Т т Ћ ћ   У у Ў ў Ф ф Х х   Ц ц Ч ч Џ џ Ш ш   Щ щ Ъ ъ Ы ы Ь ь   Э э Ю ю Я я   ́
IPA: t̪ d̪ ʈ ɖ ɟ ɡ ɢ ʡ ʔ   ɸ β θ ð ʃ ʒ ɕ ʑ ʂ ʐ ç ʝ ɣ χ ʁ ħ ʕ ʜ ʢ ɦ   ɱ ɳ ɲ ŋ ɴ   ʋ ɹ ɻ ɰ   ʙ ⱱ ʀ ɾ ɽ   ɫ ɬ ɮ ɺ ɭ ʎ ʟ   ɥ ʍ ɧ   ʼ   ɓ ɗ ʄ ɠ ʛ   ʘ ǀ ǃ ǂ ǁ   ɨ ʉ ɯ   ɪ ʏ ʊ   ø ɘ ɵ ɤ   ə ɚ   ɛ œ ɜ ɝ ɞ ʌ ɔ   æ   ɐ ɶ ɑ ɒ   ʰ ʱ ʷ ʲ ˠ ˤ ⁿ ˡ   ˈ ˌ ː ˑ ̪   {{IPA|}}