We shall state and prove a very important result in Real Analysis called the Weirstrass’ M-Test. The statement of the theorem is given below.
Let be a sequence of positive real numbers such that for , . If the series is convergent, then is uniformly convergent on .
We prove the result as follows:
If , we have the relation,
Since is convergent so there exists some such that,
The above relations imply,
It is thus clear now that is uniformly convergent on .
There are other variants of this result, which is left for a later post.