How to grep paragraph?

Hi, I have A file like this:

>Contig1
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEATATATAT
>Contig2
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTCCCA
ATCCCAAATATATAT
>Contig3
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTCCCA
ATCCCAAATAAATTTTTTCCCAATAT
ATGATATATATAEAATTTTTAATTTTT
TCCCAATCCCAAATATATATATATAT
AATATATATA
>contig1 length=130 dkfldf
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTC
>contig2 length=500 ldjfdlfjfdls
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTCCCA
ATCCCAAATAAATTTTTTCCCAATAT
ATGATATATATAEAATTTTTAATTTTT
TCCCAATCCCAAATATATATATATAT
AATATATATA
....

If i have B file like this:

Contig2
Contig3
contig2
....

How could i get a C file like this:

>Contig2
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTCCCA
ATCCCAAATATATAT
>Contig3
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTCCCA
ATCCCAAATAAATTTTTTCCCAATAT
ATGATATATATAEAATTTTTAATTTTT
TCCCAATCCCAAATATATATATATAT
AATATATATA
>contig2 length=500 ldjfdlfjfdls
AAAAAAATTTTTTCCCAATATATGAT
ATATATAEAATTTTTAATTTTTTCCCA
ATCCCAAATAAATTTTTTCCCAATAT
ATGATATATATAEAATTTTTAATTTTT
TCCCAATCCCAAATATATATATATAT
AATATATATA

Thanks!!!

awk 'END {
  if (substr(f, 2) in idx)
    print r
  }
NR == FNR {
  idx[$0]; next
  }
/^>/ {
  if (length(r)) {
    if (substr(f, 2) in idx)
      print r
    r = x; f = $1      
    }
  }  
{
  r = r ? r ORS $0 : $0
  }' file_b file_a

Thanks, but it seems not work.

You're welcome! Could you be more specific? Do you get wrong output? No output at all?