When I was young, I like the Little Mermaid in cartoon. Beautiful mermaid is really exist in the world?

No.

No.
Sailors in the older days went out into the ocean for long periods without the right food and social contact so in other words, they went mad and started drinking.
When they came back from their trip, they started telling stories to their friends about what they saw.
All alone in their boat, an unknown human sized creature would swim up to them.
The friendly creature can also be seen sitting on a rock.
Sailors even drew pictures of what they saw.
They described it as a young woman with a fish bottom half of the body.
It was like their dream lady with long hair and luring eyes.
They could sing too.
Their imaginations ended up being SEA COWS or DUGONGS.

Yes, I think mermaids exist or existed at one time.

There is no proof one way or the other, there is many forms of life not yet discovered in the oceans.

At this time there is no proof of them