Very informally, a sequence converges when there is a point, called the "limit", and the terms in the sequence get and remain as close as you want to this limit. Consider it a game: The "other" specifies …

Pointwise convergence means at every point the sequence of functions has its own speed of convergence (that can be very fast at some points and very very very very slow at others). Imagine …

Feb 17, 2025 · I think it is common to define divergence simply as non-convergence. But authors are free to define divergence to excluded bounded sequences (oscillatory sequences and chaotic …

Aug 7, 2017 · But I find it to be confusing, as from the above definition of convergence, $1/n$ turns out to be a Cauchy sequence. But wikipedia seems to be providing a different definition of convergence …

Exact definition of convergence Ask Question Asked 14 years, 1 month ago Modified 14 years, 1 month ago

The convergence is a property of the "tail". The math is just saying (in technical language) what you intuitively know: that by going far enough out into the tail of the sequence, you can guarantee that …

In proving convergence of sequences using this definition: A sequence $ (𝑎_𝑛 )$ converges to $𝑐$ if for every $\epsilon>0$, there exists an index $𝑁$ so that, for all $𝑘≥𝑁$,

Here is a very intuitive definition of convergence almost anywhere from ProofWiki: Sequence of function $(f_n)_{n \\in \\mathbb N}$ is said to converge almost everywhere on $ D$ to $ f$ if and only...

but I have no idea on an intuitive level how this defines convergence on a topological space. It seems to me by definition of convergence, you look at every open neighborhood of the convergence point. …

Feb 5, 2017 · I'm reading about the definition of exponential function on $\\Bbb R$ and I came across a definition of pointwise convergence which I don't understand: We say that a sequence of functions …