<transcript><audio_68>where k is 0 1 etc i minus 1 right because if k is i minus 1 so it's i minus 1 plus 1</transcript><transcript><audio_69>and minus 1 it will be minus i right so this is a product of these but look every this one member of this</transcript><transcript><audio_70>product is k is exactly the same now n minus k divided by n is greater than n minus k minus 1 divided by n minus 1.</transcript><transcript><audio_71>now why is this obvious well because this is equal to 1 minus k over n right and this equals to 1 minus k divided by n minus 1 right</transcript><transcript><audio_72>so from 1 we subtract this and from 1 we subtract this this is less than this because the numerators are the same denominator this is bigger denominator so the ratio is smaller so we are subtracting a smaller number so the</transcript><transcript><audio_73>result would be bigger than this one so that's why this is bigger and that's why this is bigger than this one and this y and that's why the product of these is greater than the product and everything is going back to uh original</transcript><transcript><audio_74>um inequality so that's how i have proven that my sequence is monotonically increasing now since it's monotonically increasing</transcript><transcript><audio_75>and it's bounded from the up by number three and it's actually bounded on the bottom by number two</transcript> where k is 0 1 etc i minus 1 right because if k is i minus 1 so it's i minus 1 plus 1 and minus 1 it will be minus i right so this is a product of these but look every this one member of this product is k is exactly the same now n minus k divided by n is greater than n minus k minus 1 divided by n minus 1. now why is this obvious well because this is equal to 1 minus k over n right and this equals to 1 minus k divided by n minus 1 right so from 1 we subtract this and from 1 we subtract this this is less than this because the numerators are the same denominator this is bigger denominator so the ratio is smaller so we are subtracting a smaller number so the result would be bigger than this one so that's why this is bigger and that's why this is bigger than this one and this y and that's why the product of these is greater than the product and everything is going back to uh original um inequality so that's how i have proven that my sequence is monotonically increasing now since it's monotonically increasing and it's bounded from the up by number three and it's actually bounded on the bottom by number two
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