Intro to Subsequences | Real Analysis
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- čas přidán 26. 08. 2024
- What are subsequences in real analysis? In today's lesson we'll define subsequences, and see examples and nonexamples of subsequences. We can learn a lot about a sequence by studying its subsequence, so let's talk about it!
If (a_n) is a sequence, we can denote a subsequence of (a_n) as (a_{n_k}). Here, {n_k} is itself an increasing sequence of positive integers, and you can think of it as a way of picking out the terms of (a_n) that will make up our subsequence. For example, suppose n_k = 2k-1, so that n_1 = 1, n_2 = 3, n_3 = 5, and so on. Then the subsequence (a_{n_k}) consists of the n_1th term of a_n, the n_2th term, the n_3th term, and so on. In this case, that means (a_{n_k}) consists of the n_1 = 1st term of (a_n), then the n_2 = 3rd term of (a_n), then the n_2 = 5th term of (a_n), and so on. The n_k values are picking out the terms of (a_n) to make up our subsequence.
This may sound complicated, but the idea is very simple. A subsequence of (a_n) is a sequence of terms, all from (a_n), where each term in the subsequence is further along in the original sequence. This means, for example, (a_2, a_3, a_5, a_7, a_11, ... ) could be a subsequence of (a_n), but (a_2, a_3, a_3, a_1, a_5, ...) is not, because this repeats a_3, and it goes from a_3 to a_1, which is moving backwards in the original sequence, which isn't allowed for a subsequence. As a result of how subsequences are defined, the kth term of a subsequence is at least k terms along in the original sequence.
Also, every sequence is a subsequence of itself.
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You are probably the best person for real analysis on CZcams.
Thank you so much for this video. The idea of subsequence made since to me, but the notation kept throwing me off. It is much clearer now.
really good pace of speech and repeating when necessary - good job! :)
Thanks a lot, Nativ! If you're looking for more real analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
@@WrathofMath huehue
This is very rigorous explanation 💯👌. Thank you 😊
Glad to help!
thanks, until now the internet failed to explain this :)
My pleasure, so glad it helped! The details are pretty simple, but it's very important to understand them all! If you haven't already, I hope you'll check out my Real Analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Many more lessons to come!
they definitely are tricky, but you explained it very nicely thnx
ive one doubt regarding the subsequences it will be very helpful if you can clear that...at 12:47 you mentioned nk >= k but what if i start my subsequence from second term of the parent sequence and then follow the rule that n1>n2>n3... by this my 1st term of subsequence is the 2nd term of parent sequence hence k = 2 and nk = 1 which contradicts the above inequality...does it mean that every subsequence should start with the very first term of the parent sequence.
Very helpful. Thank you so much for taking the time to share this video. On another note, you sound a lot like Sheldon Cooper.
Glad it helped, thanks for watching! And I have heard that before, I don't know if it's just because I'm talking about math or you mean my tone of voice sounds like him - I haven't really watched the show so I am not qualified to give my opinion haha! If you're looking for more real analysis, be sure to check out my real analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Many more lessons to come, let me know if you have any video requests!
your explanation is super sweet.
Thank you, Thanh! Glad it was helpful, and if you're looking for more real analysis, check out my real analysis playlist: czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Many more analysis lessons to come - let me know if you ever have any video requests!
Very helpful and detailed..
Glad it was helpful, thanks for watching!
Another great video, thank you again sir
Thaks you!!! (0,2,0,2,…,0,2,…) is a subsequence of (2,0,2,0,…,2,0,…)?
This is gold!
Thank you! If you're looking for more analysis, check out my playlist! czcams.com/play/PLztBpqftvzxWo4HxUYV58ENhxHV32Wxli.html
Pls, iam in a bad need to this vedio ..may u pls make the transcript in English 🥺
my question is why a sequence is considered a subsequence of itself
if a sequence represents the whole thing
then how can in logic a whole thing be part of itself
how can the sub/part equals the whole ?
i tried to understand it but i couldnt
can anyone help me with some examples to understand it fully ?
because it is defined to be. don't know if this helps but it's similar to how a set is a subset of itself.
also, being "a subsequence of a sequence" should not necessarily mean "a part of a sequence." i think the proper term would be a proper subsequence.
wow! very well explained! thanks you!
Thank you so much sir 😊❤
Great video, however I think u meant nk
yeah cuz if we take n_1=10 we are done
idk what was i thinking but was checking my notes today and came back to watch this vid to see if it was really n_k
Interested
at 11:29 shouldnt n5=7 be n4=7?
Thanx man 👍👍
Clear!
Perfect!