let X be non empty set ; :: thesis: for f being Functional_Sequence of X,ExtREAL
for x being Element of X st x in dom (lim f) & f # x is convergent holds
( (lim f) . x = (lim_sup f) . x & (lim f) . x = (lim_inf f) . x )
let f be Functional_Sequence of X,ExtREAL ; :: thesis: for x being Element of X st x in dom (lim f) & f # x is convergent holds
( (lim f) . x = (lim_sup f) . x & (lim f) . x = (lim_inf f) . x )
let x be Element of X; :: thesis: ( x in dom (lim f) & f # x is convergent implies ( (lim f) . x = (lim_sup f) . x & (lim f) . x = (lim_inf f) . x ) )
assume A1: ( x in dom (lim f) & f # x is convergent ) ; :: thesis: ( (lim f) . x = (lim_sup f) . x & (lim f) . x = (lim_inf f) . x )
then x in dom (f . 0 ) by Def10;
then ( x in dom (lim_sup f) & x in dom (lim_inf f) ) by Def8, Def9;
then A2: ( (lim_sup f) . x = lim_sup (f # x) & (lim_inf f) . x = lim_inf (f # x) ) by Def8, Def9;
( lim (f # x) = lim_sup (f # x) & lim (f # x) = lim_inf (f # x) ) by A1, RINFSUP2:41;
hence ( (lim f) . x = (lim_sup f) . x & (lim f) . x = (lim_inf f) . x ) by A1, A2, Def10; :: thesis: verum