theorem :: LIMFUNC1:45

for f being PartFunc of REAL,REAL holds
( f is convergent_in-infty iff ( ( for r being Real ex g being Real st
( g < r & g in dom f ) ) & ex g being Real st
for g1 being Real st 0 < g1 holds
ex r being Real st
for r1 being Real st r1 < r & r1 in dom f holds
|.((f . r1) - g).| < g1 ) )