# Tag Archives: convergence

## Topology: Nets and Points of Accumulation

Recall that a sequence in a topological space X converges to a in X if the function f : N* → X which takes is continuous at . Unrolling the definition, it means that for any open subset U of X containing a, the set contains (N, ∞] for some N. In … Continue reading

## Topology: Limits and Convergence

Following what we did for real analysis, we have the following definition of limits. Definition of Limits. Let X, Y be topological spaces and . If  f : X-{a} → Y is a function, then we write if the function: is … Continue reading

## Basic Analysis: Uniform Convergence

Once again, let be a subset. Suppose we now have a sequence of functions , where n = 1, 2, 3, … , such that for each x in D, the sequence converges to some real value. We’ll denote this value by f(x), thus … Continue reading

Posted in Notes | | 1 Comment

## Basic Analysis: Sequence Convergence (4)

In this article, we’ll consider the convergence of an infinite sum: . We call this sum an infinite series. Let be the partial sums of the series. Definition. We say that is L (resp. ∞, -∞) if the partial sums converge to … Continue reading

## Basic Analysis: Sequence Convergence (3)

So far, we’ve been considering the case where a sequence converges to a real number L. It’s also possible for a sequence to approach +∞ or -∞. The infinity symbol “∞” should be thought of as a convenient symbol instead of … Continue reading