<transcript><audio_30>all right so how can i prove that this thing has a limit a little bit more rigorously than what i did in the previous lecture the previous lecture was kind of the intuitive approach now i will make it a little bit more</transcript><transcript><audio_31>rigorous now um there was a theorem which i have proved in one of the previous lectures dedicated to limits</transcript><transcript><audio_32>which is monotonically increasing and bound from above let me just do it graphically so this is my boundary a</transcript><transcript><audio_33>and my sequence is monotonically increasing and always less than eight so my points are</transcript><transcript><audio_34>so the theorem says that if my sequence is monotonically increasing and always bounded on the top by some constant a then there is a limit and this limit is actually the least</transcript><transcript><audio_35>for this particular sequence which is called so what my point right now is that i will use this theorem</transcript><transcript><audio_36>which states that if it's bounded and monotonically increasing then there is a limit and that's what actually i have to prove here that there is a limit which i can call the number e so if i will prove that this is monotonically increasing</transcript><transcript><audio_37>sequence i already know it's bounded from above by the number three that would be sufficient to prove that there is a limit and if there is a limit i can call it anything i want including</transcript><transcript><audio_38>this beautiful letter e which mathematicians like very much all right so all i have to prove again</transcript> all right so how can i prove that this thing has a limit a little bit more rigorously than what i did in the previous lecture the previous lecture was kind of the intuitive approach now i will make it a little bit more rigorous now um there was a theorem which i have proved in one of the previous lectures dedicated to limits which is monotonically increasing and bound from above let me just do it graphically so this is my boundary a and my sequence is monotonically increasing and always less than eight so my points are so the theorem says that if my sequence is monotonically increasing and always bounded on the top by some constant a then there is a limit and this limit is actually the least for this particular sequence which is called so what my point right now is that i will use this theorem which states that if it's bounded and monotonically increasing then there is a limit and that's what actually i have to prove here that there is a limit which i can call the number e so if i will prove that this is monotonically increasing sequence i already know it's bounded from above by the number three that would be sufficient to prove that there is a limit and if there is a limit i can call it anything i want including this beautiful letter e which mathematicians like very much all right so all i have to prove again
{"source": "grass-yt-cc-by.0d_4AewmTJo|shard_10014_20|create_multimodal_data.extract_yt_and_generate_transcripts.50|shard_10014|create_multimodal_data.generate_captions.54|shard_10014|create_multimodal_data.generate_captions.45", "params": {"id": "0d_4AewmTJo", "channel_id": "UCNwHL7lffNALP62igvVhdkA", "duration": 1563, "view_count": 3885, "age_limit": 0, "like_count": 37, "channel_follower_count": 2890, "license": "Creative Commons Attribution license (reuse allowed)", "uploader_id": "@ZorShekhtman", "yt_tar_file": "/leonardo_work/EUHPC_E03_068/audio/fixed_download/grass-yt-cc-by//shard_10014/0d_4AewmTJo.tar.gz"}}