You have visitors at your house. You offered them something to drink. They asked what you have, so you answer:
Well, let me see... I have water? Sprite? beer?
One reason for saying "well" at the beginning of a sentence is to show that you're unsure about something.
This can happen when someone asks you a question that you don't immediately know the answer to:
A: What are you doing this weekend?
B: Well, let's see... Oh! I'm having lunch with Gavin on Saturday.
It can also happen when you're trying to say something nice, even though you're thinking things that aren't nice.
A: What do you think of Lana?
B: Well... she's... always very punctual.
You shouldn't use "well" in this way in written English.
You say "let me see" when you're trying to remember something, or when you're checking on a computer or a piece of paper for information. For example, when you're calling to make an appointment for a dentist, you may hear:
A: Would it be possible to make an appointment for some time this week?
B: Let me see... we have an opening on Tuesday at noon. Would that work?
The receptionist says "let me see" while he searches the computer for available times.
When you're asking people to choose from among several options, you pronounce each choice as a question:
What is your problem? Are you sleepy? Hungry? Do you need your diaper changed?
You can use question intonation for each choice even if the sentence isn't really a question. The phrase at top is an example of this.
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