Final answer:
To show that the sequence 1/n is convergent with a limit of zero, we use the definition of the limit of a sequence. According to the definition, a sequence converges to a limit L if for every positive number ε, there exists a positive integer N, such that for all n greater than or equal to N, the terms of the sequence are within ε of L. By choosing an arbitrary positive ε and finding a value of N that satisfies the condition, we can show that the sequence 1/n converges to zero.
Step-by-step explanation:
To show that the sequence 1/n is convergent with a limit of zero, we need to use the definition of the limit of a sequence.
According to the definition, a sequence converges to a limit L if for every positive number ε, there exists a positive integer N, such that for all n greater than or equal to N, the terms of the sequence are within ε of L.
In the case of the sequence 1/n, we want to show that for any ε greater than zero, there exists an integer N, such that for all n greater than or equal to N, |1/n - 0| < ε.
Let's choose an arbitrary positive ε and find a value of N that satisfies the condition. Let's say we choose ε = 0.01.
In order for |1/n - 0| < 0.01 to hold, we need n to be greater than or equal to 100. This means that if we choose N = 100, for every n greater than or equal to 100, |1/n - 0| < 0.01.
Therefore, we have shown that for any positive ε, there exists a positive integer N, such that for all n greater than or equal to N, |1/n - 0| < ε. This proves that the sequence 1/n converges to zero.