Prove that if an is a convergent sequence then an is a conve

Prove that if {a_n} is a convergent sequence, then {|a_n| is a convergent sequence. Is the converse true?

Solution

Yes

{an} is convergent. Let its limit be L

So for any e>0 there exists, N so that for all n>N

|an-L|<e

||an|-|L||<|an-L|<e for all n>L

HEnce, {|an|} converges to |L|