§l. Convergent sequences and series
97
lim f(n)jg(n) = apjbp.
n-+oo
The proof is the same as above: one divides the two members of the fraction
by n P and remarks that ljn and all its powers tend to 0 as n increases
indefinitely.
Beyond these rules of algebraic calculation, there is another simple operation which transforms one convergent sequence into another. Let (un) be a
sequence which converges to U and let f be a scalar function defined on a
neighbourhood of u, except perhaps at u, and such that f(x) tends to a limit
v when x tends to u. Then f(un ) tends to v. For all r > 0, there is indeed an
r' > ° such that Ix - ul < r' implies If(x) - vi < rj then one has IUn - ul < r'
for n large, whence If(un ) - vi < r for n large.
If in particular a function f is continuous at a point a, then
(8.2)
lim Un = a ==? lim f(u n ) = f(a),
a fundamental result even though almost trivial (i.e. following directly from
the definitions).