reserve n,m,k for Nat;
reserve a,p,r for Real;
reserve s,s1,s2,s3 for Real_Sequence;

theorem Th1:
  0<a & a<1 & (for n holds s.n=a to_power (n+1)) implies
  s is convergent & lim s = 0
proof
  assume that
A1: 0<a and
A2: a<1 and
A3: for n holds s.n=a to_power (n+1);
  set r = 1/a - 1;
  1/a > 1/1 by A1,A2,XREAL_1:88;
  then
A4: r > 0 by XREAL_1:50;
A5: for p be Real st 0 < p ex m st for n st m<=n holds |.s.n - 0.| < p
  proof
    let p be Real;
A6: 1/(p * r) <= [\1/(p * r)/] + 1 by INT_1:29;
    assume
A7: 0<p;
    then p * r > 0 by A4;
    then 1/(p * r) > 0;
    then 0 < [\1/(p * r)/] + 1 by A6;
    then [\1/(p * r)/] is Element of NAT by INT_1:3,7;
    then consider m such that
A8: m = [\1/(p * r)/];
    take m;
A9: 1/(p * r) = 1/p/r by XCMPLX_1:78;
    now
A10:  [\1/(p * r)/] > 1/(p * r) -1 by INT_1:def 6;
      let n;
      assume m<=n;
      then n > 1/(p * r) - 1 by A8,A10,XXREAL_0:2;
      then n + 1 > 1/p/r by A9,XREAL_1:19;
      then
A11:  (n+1) * r > 1/p by A4,XREAL_1:77;
      0 + (n+1) * r < 1 + (n+1) * r by XREAL_1:6;
      then
A12:  1/p < 1 + (n+1) * r by A11,XXREAL_0:2;
A13:  (1+r) to_power (n+1) = 1 to_power (n+1) / a to_power (n+1) by A1,POWER:31
        .= 1 / a to_power (n+1);
      a to_power (n+1) > 0 by A1,POWER:34;
      then
A14:  |.a to_power (n+1).| = a to_power (n+1) by ABSVALUE:def 1;
      1 + (n+1) * r <= (1 + r) to_power (n + 1) by A4,POWER:49;
      then 1/p < 1/ a to_power (n+1) by A13,A12,XXREAL_0:2;
      then |.a to_power (n+1).| < p by A7,A14,XREAL_1:91;
      hence |.s.n - 0.| < p by A3;
    end;
    hence thesis;
  end;
  hence s is convergent by SEQ_2:def 6;
  hence thesis by A5,SEQ_2:def 7;
end;
