True/False regarding Delta Neighborhoods

[RJLiberator]

Homework Statement

True/False: If a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any smaller positive delta will also suffice.

Homework Equations

The Attempt at a Solution

The submitted solution is as follows:

However, when I read this solution, I note that 0 < delta_1 < delta_2.

The submitter goes on to start with delta_1 to then show that delta 2 holds. Didn't the submitter show that if a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any LARGER positive delta will also suffice? This is not what the question asks.

 

[Mark44]

Homework Statement

True/False: If a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any smaller positive delta will also suffice.

Homework Equations

The Attempt at a Solution

The submitted solution is as follows:
View attachment 113966

However, when I read this solution, I note that 0 < delta_1 < delta_2.

The submitter goes on to start with delta_1 to then show that delta 2 holds. Didn't the submitter show that if a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any LARGER positive delta will also suffice? This is not what the question asks.

No, it is not saying that a larger delta works. Here's what I think is going on. A challenge value of $\epsilon > 0$ has been given, which is answered by a value of $\delta_2$. In the image, a smaller value of $\delta_1$ is then selected. Now, if $|x - c | < \delta_2$ it will also be true (almost trivially) that $|x - c | < \delta_1$, which in turn implies that $|f(x) - f(c)| < \epsilon$

 

[PeroK]

Homework Statement

True/False: If a particular delta has been constructed as a suitable response to a particular epsilon challenge, then any smaller positive delta will also suffice.

There are two aspects to limits: an understanding of what you are trying to do; and, the nitty-gritty manipulation of epsilons and deltas etc.

In this case, the understanding should be clear: finding a delta means you are "close enough" to a point and if you reduce the delta you are "even closer". While, increasing the delta means you are "further away".

You really shouldn't be having any trouble seeing this.