let s be Real_Sequence; :: thesis: ( ( for n being Nat st n >= 1 holds
( s . n = sqrt (n * (n + 1)) & s . 0 = 0 ) ) implies for n being Nat st n >= 1 holds
(Partial_Sums s) . n > (n * (n + 1)) / 2 )
defpred S₁[ Nat] means (Partial_Sums s) . $1 > ($1 * ($1 + 1)) / 2;
assume A1: for n being Nat st n >= 1 holds
( s . n = sqrt (n * (n + 1)) & s . 0 = 0 ) ; :: thesis: for n being Nat st n >= 1 holds
(Partial_Sums s) . n > (n * (n + 1)) / 2
A2: for n being Nat st n >= 1 & S₁[n] holds
S₁[n + 1]

proof

let n be Nat; :: thesis: ( n >= 1 & S₁[n] implies S₁[n + 1] )
assume that
A3: n >= 1 and
A4: (Partial_Sums s) . n > (n * (n + 1)) / 2 ; :: thesis: S₁[n + 1]
n + 1 >= 1 + 1 by A3, XREAL_1:7;
then n + 1 >= 1 by XXREAL_0:2;
then A5: s . (n + 1) = sqrt ((n + 1) * ((n + 1) + 1)) by A1;
n + 2 > n + 1 by XREAL_1:8;
then (n + 2) * (n + 1) > (n + 1) ^2 by XREAL_1:68;
then sqrt ((n + 2) * (n + 1)) > sqrt ((n + 1) ^2) by SQUARE_1:27;
then A6: sqrt ((n + 2) * (n + 1)) > n + 1 by SQUARE_1:22;
((Partial_Sums s) . n) + ((sqrt ((n + 1) * (n + 2))) - (((n + 1) * (n + 2)) / 2)) > ((n * (n + 1)) / 2) + ((sqrt ((n + 1) * (n + 2))) - (((n + 1) * (n + 2)) / 2)) by A4, XREAL_1:8;
then (((Partial_Sums s) . n) + (sqrt ((n + 1) * (n + 2)))) - (((n + 1) * (n + 2)) / 2) > (sqrt ((n + 1) * (n + 2))) - (n + 1) ;
then (((Partial_Sums s) . n) + (sqrt ((n + 1) * (n + 2)))) - (((n + 1) * (n + 2)) / 2) > 0 by A6, XREAL_1:50;
then A7: ((((Partial_Sums s) . n) + (sqrt ((n + 1) * (n + 2)))) - (((n + 1) * (n + 2)) / 2)) + (((n + 1) * (n + 2)) / 2) > 0 + (((n + 1) * (n + 2)) / 2) by XREAL_1:8;
thus S₁[n + 1] by A5, A7, SERIES_1:def 1; :: thesis: verum

end;

A8: (Partial_Sums s) . 1 = (Partial_Sums s) . (0 + 1)
.= ((Partial_Sums s) . 0) + (s . 1) by SERIES_1:def 1
.= (s . 0) + (s . 1) by SERIES_1:def 1
.= 0 + (s . 1) by A1
.= sqrt (1 * (1 + 1)) by A1
.= sqrt 2 ;
then ((Partial_Sums s) . 1) ^2 = 2 by SQUARE_1:def 2;
then sqrt (((Partial_Sums s) . 1) ^2) > sqrt (((1 * (1 + 1)) / 2) ^2) by SQUARE_1:27;
then A9: sqrt (((Partial_Sums s) . 1) ^2) > (1 * (1 + 1)) / 2 ;
(Partial_Sums s) . 1 > 0 by A8, SQUARE_1:25;
then A10: S₁[1] by A9, SQUARE_1:22;
for n being Nat st n >= 1 holds
S₁[n] from NAT_1:sch 8(A10, A2);
hence for n being Nat st n >= 1 holds
(Partial_Sums s) . n > (n * (n + 1)) / 2 ; :: thesis: verum