My professor vaguely defined a cluster point and I'm a bit confused on the difference between limit point and cluster point.

He said that

" Suppose a_n converges to alpha.

Sequence {a_n} has a cluster point, if a subsequence of a_n converges to alpha.

If a is a cluster point, it is a limit point."

Can you clarify the difference between cluster point and limit point.

I am currently in a introductory analysis class.

Thank you.