You might have noticed that the phrases for greeting and parting ended in -ㅔ요, as with:
안녕하세요
The phrases when introducing oneself to people for the first time all end in -ㅂ니다, as in
뵙겠습니다
The -ㅂ니다 form is used when meeting someone for the first time because it is considered to be more formal than the -ㅔ요 form. You need not worry about being unintentionally rude when using an -ㅔ요 form; it is informal but still considered polite.
This topic of levels of politeness and formality is deeply knitted into the Korean language, and is covered in great detail in most Korean grammar books.
Introductions
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