# Showing that there are particular sequences of functions that converge

• jdinatale
In summary, the speaker is discussing how to prove that \phi_n converges to f and how to show that \psi_n is decreasing. The other person agrees that there seems to be a missing argument. They suggest defining \tilde{\phi_n}(x) to show convergence and prove its properties.
jdinatale

That seems like a valid argument for showing that $\phi_n$ converges to f, but I'm not sure how to show it's increasing. And as far $\psi_n$, converges, well I imagine that I'd use a similar argument, but I'm still not sure how to show it's decreasing.

I agree it seems to be missing.

Since $\phi_n$ is smaller than f, you can define
$$\tilde{\phi_n}(x) = \max_{j=1,...,n} \phi_k(x)$$
and this will still be within 1/n of f everywhere, and is increasing in n, and each $\tilde{\phi_n}$ is a simple function. Exercise to prove all these properties.

## 1. What does it mean for a sequence of functions to converge?

Convergence of a sequence of functions means that as the number of terms in the sequence increases, the functions approach a single limiting function. In other words, the values of the functions get closer and closer to each other as the sequence progresses.

## 2. How do you prove that a sequence of functions converges?

There are several ways to prove that a sequence of functions converges. One method is to show that the limit of the sequence exists and is equal to the limiting function. Another method is to use the definition of convergence, which states that for any small number ε, there exists a value of n such that the difference between the limiting function and the nth term of the sequence is less than ε.

## 3. Can a sequence of functions converge to more than one limiting function?

No, a sequence of functions can only converge to one limiting function. This is because the definition of convergence requires that the values of the functions get closer and closer to each other, and if there were multiple limiting functions, the values would not necessarily get closer to each other.

## 4. What are some examples of sequences of functions that converge?

One example is the sequence of functions f_n(x) = x^n, where x is a real number between -1 and 1. This sequence converges to the limiting function f(x) = 0 for x ≠ 1, and f(x) = 1 for x = 1. Another example is the sequence of functions g_n(x) = 1/n, which converges to the limiting function g(x) = 0 for all values of x.

## 5. Are there any properties that a sequence of functions must have in order to converge?

Yes, there are several properties that a sequence of functions must have in order to converge. These include being defined on a closed interval, being continuous, having a finite limit, and satisfying certain convergence criteria such as the Cauchy criterion or the Monotone Convergence Theorem. The properties required for convergence may vary depending on the specific sequence of functions being considered.

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