Real Analysis 14 | Heine-Borel Theorem [dark version]

[@michaeloppong1240]

Well explained video. Thanks

[@brightsideofmaths]

Thanks :)

[@KurdaHussein]

where can I find a proof of that theorem for Rⁿ ?

[@brightsideofmaths]

It's literally the same proof as the one shown here.

[@sub2jkoozi820]

Just wanted to say how great your videos are! You are a great help to many, thanks :)

[@sounakroy1933]

how does divergence to infinity mean that the set has not accumulation values?

[@brightsideofmaths]

"Divergent to infinity" essentially means "convergent to infinity"

[@sounakroy1933]

@@brightsideofmaths i am confused with the part where you say since |an| > n for all n in N for some sequence (an) in A it then imples that A has no accumulation values.

[@brightsideofmaths]

​@@sounakroy1933 What does accumulation value mean?

[@sounakroy1933]

@@brightsideofmaths Accumulation values are generalisation of limits. A point a in R us called accumulation value of sequence (an) if there is a subsequence (ank) with limit a. If this arbitrary sequence is a convergent sequence then we know that limit f sequence and subsequence both converge to a. Right?

[@sounakroy1933]

@@brightsideofmaths so an unbounded sequence will diverge to infinity or - infinity. And terms of any subsequence won't converge right?