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A Theorem and Proof

Theorem:

Suppose that f is a continuous function and that 

  Lim    p(n) = L
n --> oo

Then 

  Lim    f(p(n)) = f(L)
n --> oo

Proof:

Given a permissible error epsilon we must find an N such that for every n >= N 

|f(p(n)) - f(L)| < epsilon.

Since f is comntinuous there is a tolerance delta such that if 

|x - L| < delta

then 

|f(x) - f(L)| < epsilon

and since 

  Lim    p(n) = L
n --> oo

there is an N such that if n >= N then 

|p(n) - L| < delta

which implies that 

|f(p(n)) - f(L)| < epsilon

and completes the proof.

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Copyright c 1995 by PWS Publishing Company, a division of International Thomson Publishing Inc. Comments to Frank Wattenberg, Department of Mathematics, Carroll College, Helena, MT 59625.