Prove that if every PROPER sub sequence of a sequence is convergent , then the sequence is convergent

Prove that if every PROPER sub sequence of a sequence is convergent , then the sequence is convergent. ( By proper, i mean all the sub sequences excluding the main sequence )

$Attempt$ : Since all proper sub sequences are convergent

=> Each of them must be bounded And Each of them must have their unique limit points

Since EACH of the sub sequences are bounded, then, the main sequence must be also bounded

=> By Bolzanno Weirstrass Theorem, the bounded main sequence has a limit point. ......(1)

Now, If i prove that the bounded main sequence converges to a unique limit point, then , I can prove that the sequence is convergent.

To do this, If i prove that all the sub sequences converge to one limit point l only, then from (1) we can conclude that the unique limit of the sequence is l .

How do i do this? Any directions?

Thanks