For each of the following prove or give a counter example A

For each of the following prove or give a counter example

A) let Sn be a sequence that converges to s (Sn->s), then absolute value (|Sn|) converges to |s|.

B) If |Sn| is convergent, then Sn is convergent.

C) Lim Sn = 0 iff lim |Sn| = 0

Solution

a) True

When Sn converges to s say a negative number, if Sn is positive, then absolute value of Sn again converges to s.

If Sn is negative, then |Sn| =-Sn hence converges to s i.e. |-s|=s

b) True.

If |Sn| is convergent then Sn would be a finite number either positive or negative.

Hence Sn has to be convergent

c) Lim Sn =0 means

left limit = right limit = 0

|Sn|= Sn if Sn is positive hence limit =0

If Sn is negative |Sn| = -Sn converges -0 =0

Hence true .