Advanced Calculus fi..

Chapter 6 Infinite Series 381Every sequence will then possess upper and lower limits. It should be noted thatwhen lim s, = -oo, the upper and lower limits are both -oo. In general, the lowerlimit cannot exceed the upper limit.THEOREM 3 If the sequence s, converges to s, then-lim sn = !@ s, = s;n+oc n+ccconversely, if upper and lower limits are equal and finite, the sequence converges. 1The first part of the theorem follows from the definition of convergence; forthe limit s satisfies all conditions for both upper and lower limits. If the upper andlower limits are both finite and equal to s, the sequence is necessarily bounded, forotherwise the upper limit would be +oo or the lower limit would be -00. If s, doesnot converge to s, then for some E there are infinitely many members of the sequencesuch that Is, - s I > E . One can then proceed as in the proof of Theorem 1 to producea limiting value k other than s. This contradicts the fact that s is both the largest and .least limiting value.We first point out connections between sequences and continuity of functions. 3msTHEOREM 4xo, thenLet y = f (x) be defined for a 5 x 5 b. If f (x) is continuous ati tfor every sequence x, in the interval converging to xo. Similarly, iff (x, y) is definedin a domain D and is continuous at (xO, yo), thenfor all sequences x, , y, such that (x, , yn) is in D and x, converges to xo, y, convergesto Yo.We prove the statement for f (x), the proof for a function of two (or more)variables being similar. Given E > 0, a 8 can be found such that If(x)- f (xo)l < E forIx -xo I < 8; this is simply the definition of continuity. One can then choose N so thatIx, -xol < 6 for n > N, sincex, converges toxo. Accordingly, I f (x,,) - f (xo)l < 6for n > N; that is, f (x,) converges to f (xo).We remark that there is a converse to Theorem 4: If f (x) has the property thatf (x,,) converges to f (xo) for every sequence x, that converges to xo, then f (x) mustbe continuous at xo (Theorem G in Section 2.23).. . - I