Hello. So our first statement here is that for every act So this upside down a means for all. So for all Acts X is less than X plus one. Well that is certainly true. Um The reason why I was kind of easy but the reason is that X plus one is always going to be one more than X. Right X plus one is always one more than X. So so yes for any X you add one to it. Well um you get a number which is bigger than that. So access all gonna be less than expose one for any X. That's true. Our next statement. This is gonna be a statement A for statement. Oops after A and then for B. Um we have our statement is there exists a natural number end? So this backwards. E means there exists um a natural number end. So there exists um an end that is a natural number. So an element of this fancy and means the natural number. So there exists an end an element of the natural numbers. In other words, there exists a natural number and um such that all prime numbers. All prime numbers are less than N. Okay, well that statement is definitely false. Um and um well, I mean basically there's a there's a proven good right here. But I mean it's simple proof. I mean, look it up if you don't know it yet, but actually the prime numbers are infinite. So um there exists an end, right? Such that all prime numbers are less than end. Well, okay, we know that the national numbers go on forever. But um the prime numbers, right, go on for ever. Remember you pick there's going to be prime numbers bigger than it. So this statement is false. And then our next statement um is going to be for every X greater than zero. So for all X greater than zero, there exists. Um there exists why? So there exists against backwards evening. There exists um a Y. Such that so such that why is greater than one over X. Um This is actually true. And the reason there is, well, we can certainly pick a number why that is larger than one over X. Um for any value of X that we picked. And our idea our statement is that for every positive number acts. So for all acts greater than zero for any positive acts, there exists a natural number end. So there exists a natural number end. So there exists an element of the natural numbers, there exists a natural number end. Um Such that one over End is going to be less than X. Well, that statement is true, that statement is true and the reason is, well, we I mean the natural numbers are just accounting numbers 1234 and so on. And since end here is in the denominator, we can make the fraction uh one over and here. Um as small as we like, by just changing end to be a larger and larger and larger number. So we can always be able to make one over end. Um less than X, no matter what we pick four X. And our four part E r statement is that for every positive epsilon, epsilon is typically denoted as a very small positive number. So for every positive epsilon, we can regulate this. As for all epsilon, epsilon is a greek letter which looks like a backward street. So for all epsilon um greater than zero. For all positive absalon. So for all epsilon greater than zero, there exists a natural number. End. So there exists again a natural number. End. So there exists an n An element of the natural numbers such that one over to to the end is going to be less than epsilon. Okay, that statement is yes, it is true. And the reason behind this is well, since then um is in the power of to write um in the denominator we can make to to the end as large as we desire, rights similar to that, we can make to to the end as large as we like. Um and therefore one over to to the end can get as small as we desire because it is to to the end gets bigger and bigger and bigger and bigger. One over to to the end get smaller and smaller and smaller and smaller. So one over to to the end can be made as small as we desire by changing end to be a larger and larger number, so therefore we can always be able to make one over to to the end less than epsilon, No matter what positive number we pick for epsilon. Take care.