The analysis problem you have presented deals with sequences, specifically the limit of a sequence. In order to solve this problem, we need to understand the definitions and properties of limits and sequences.
A sequence is a list of numbers arranged in a specific order. In this case, the sequence is denoted by t_n and is defined as the average of the first n terms of another sequence, denoted by s_n. The average is calculated by adding all the terms and dividing by the number of terms.
A limit is the value that a sequence approaches as the number of terms increases. In this case, we are interested in the limit as n approaches infinity. This means that we are looking at the behavior of the sequence as the number of terms becomes larger and larger.
Now, let's look at the first statement: if lim n-> [infinity] s_n = s. This means that as n approaches infinity, the terms of the sequence s_n get closer and closer to the value s. In other words, the terms of the sequence s_n are approaching the limit s.
Next, we have to show that if this statement is true, then lim n-> [infinity] t_n = s. This means that if the terms of the sequence s_n are approaching the limit s, then the terms of the sequence t_n, which are the averages of the terms of s_n, are also approaching the limit s.
To prove this, we can use the definition of the limit. We have to show that for any small positive number ε, there exists a positive integer N such that for all n > N, the difference between t_n and s is less than ε. In other words, we have to show that t_n is getting closer and closer to s as n approaches infinity.
Since we know that lim n-> [infinity] s_n = s, we can choose a large enough N such that for all n > N, the difference between s_n and s is less than ε/2. This means that for all n > N, we have:
|s_n - s| < ε/2
Now, let's look at the definition of t_n:
t_n = [s_1 + s_2 + ... + s_n] / n
We can rewrite this as:
t_n = [(s_1 - s) + (s_2 - s) + ... + (s_n - s) + ns] / n
Notice that
