Q: Older than I or Older than me?
A: 'Older than me' is sometimes considered informal while 'older than I' is quite archaic. Some assessment books and school teachers won't accept 'older than me' but this is not correct. If one uses 'I', then a verb should be placed after the 'I' for example, 'older than I am' Therefore, to answer your question: 'Older than me' is correct; 'Older than I am' is correct. To test, We normally say 'She is older than he IS.' or 'She is older than him,' but not 'She is older than he.'
Q: I lay down for a nap? or I lie down for a nap?
A: Correct answer is I lie down for a nap. Lie (down) is in the present tense. Past tense: lay (down): I lay down for a nap yesterday. The hen lays an egg. (present tense). The hen laid an egg. (past tense) So: do not confuse 'lie' with 'lay'.

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