let Rseq be Function of ,REAL; :: thesis: ( Rseq is P-convergent iff lim_filter ((# Rseq),) <> {} )
hereby :: thesis: ( lim_filter ((# Rseq),) <> {} implies Rseq is P-convergent )
assume Rseq is P-convergent ; :: thesis: lim_filter ((# Rseq),) <> {}
then consider x being Real such that
A1: for e being Real st 0 < e holds
ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - x).| < e ;
for m being non zero Nat ex n being Nat st
for n1, n2 being Nat st n <= n1 & n <= n2 holds
|.((Rseq . (n1,n2)) - x).| < 1 / m
proof
let m be non zero Nat; :: thesis: ex n being Nat st
for n1, n2 being Nat st n <= n1 & n <= n2 holds
|.((Rseq . (n1,n2)) - x).| < 1 / m

0 / m < 1 / m by XREAL_1:74;
hence ex n being Nat st
for n1, n2 being Nat st n <= n1 & n <= n2 holds
|.((Rseq . (n1,n2)) - x).| < 1 / m by A1; :: thesis: verum
end;
hence lim_filter ((# Rseq),) <> {} by Th58; :: thesis: verum
end;
assume lim_filter ((# Rseq),) <> {} ; :: thesis: Rseq is P-convergent
then consider p being object such that
A2: p in lim_filter ((# Rseq),) by XBOOLE_0:def 1;
reconsider p = p as Real by A2;
ex p0 being Real st
for e being Real st 0 < e holds
ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p0).| < e
proof
take p ; :: thesis: for e being Real st 0 < e holds
ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p).| < e

hereby :: thesis: verum
let e be Real; :: thesis: ( 0 < e implies ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p).| < e )

assume 0 < e ; :: thesis: ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p).| < e

then ex m being Nat st
( not m is zero & 1 / m < e ) by Th5;
then consider m0 being non zero Nat such that
not m0 is zero and
A3: 1 / m0 < e ;
consider n0 being Nat such that
A4: for n1, n2 being Nat st n0 <= n1 & n0 <= n2 holds
|.((Rseq . (n1,n2)) - p).| < 1 / m0 by ;
now :: thesis: ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p).| < e
take N = n0; :: thesis: for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p).| < e

hereby :: thesis: verum
let n, m be Nat; :: thesis: ( n >= N & m >= N implies |.((Rseq . (n,m)) - p).| < e )
assume ( n >= N & m >= N ) ; :: thesis: |.((Rseq . (n,m)) - p).| < e
then |.((Rseq . (n,m)) - p).| < 1 / m0 by A4;
hence |.((Rseq . (n,m)) - p).| < e by ; :: thesis: verum
end;
end;
hence ex N being Nat st
for n, m being Nat st n >= N & m >= N holds
|.((Rseq . (n,m)) - p).| < e ; :: thesis: verum
end;
end;
hence Rseq is P-convergent ; :: thesis: verum