Let annN be a sequence of real numbers Show that annN is con

Let (an)nN be a sequence of real numbers. Show that
(an)nN is convergent if and only if (an)nN is bounded and (an)nN possesses exactly one cluster point.

Solution

Proof in reverse

I.e consider function to be convergent

Condition for the series to be convergent is that it should be a bounded function for which series terminates to zero as n tends to infinity. And the exactly one cluster point