let seq, seq1 be Real_Sequence; :: thesis:
assume A1: seq is convergent ; :: thesis:
given k being Nat such that A2: for n being Nat st k <= n holds
seq1 . n = seq . n ; :: thesis:

A3: now :: thesis:

let p be Real; :: thesis:
assume 0 < p ; :: thesis:
then consider n1 being Nat such that
A4: for m being Nat st n1 <= m holds
|.((seq . m) - (lim seq)).| < p by A1, SEQ_2:def 7;

A5: now :: thesis:

assume A6: n1 <= k ; :: thesis:
take n = k; :: thesis:
let m be Nat; :: thesis:
assume A7: n <= m ; :: thesis:
then n1 <= m by A6, XXREAL_0:2;
then |.((seq . m) - (lim seq)).| < p by A4;
hence |.((seq1 . m) - (lim seq)).| < p by A2, A7; :: thesis:

end;

now :: thesis:

assume A8: k <= n1 ; :: thesis:
take n = n1; :: thesis:
let m be Nat; :: thesis:
assume A9: n <= m ; :: thesis:
then seq1 . m = seq . m by A2, A8, XXREAL_0:2;
hence |.((seq1 . m) - (lim seq)).| < p by A4, A9; :: thesis:

end;

hence ex n being Nat st
for m being Nat st n <= m holds
|.((seq1 . m) - (lim seq)).| < p by A5; :: thesis:

end;

seq1 is convergent by A1, A2, Th18;
hence lim seq = lim seq1 by A3, SEQ_2:def 7; :: thesis: