Question

Give an example of an unbounded sequence which has a convergent subsequence. Justify your claims

Answer 1

An = 2^(n*(-1)^n)
Basically, the n part of the power, makes the exponent grow in an unbounded way, but the (-1)^n makes the POWER alternate between negative and positive. That means the every even term of the sequence will be growing in an unbounded way, while the odd terms of the sequence will become very small fractions.
Notice the pattern
A1 = 2^-1 = (1/2)
A2 = 2^2 = 4
A3 = 1/8
A4 = 16
A5 = 1/32
A6 = 64
A7 = 1/128
...