**Eventually or Frequently?**

The two terms, “eventually” and “frequently,” stood out to me as having fairly interesting definitions when the came up some introductory courses in analysis and toplogy while I was in graduate school. Specifically, I am referring their use in the context of infinite sequences. Now to lay the foundation for the illustration I’d like to make, recall that the definition of an epsilon-neighborhood of a point, [tex]a in mathbb{R}[/tex], is the interval [tex] (a-varepsilon,a+varepsilon)[/tex], for some [tex]varepsilon[/tex]. In other words, it is the set of all points within [tex]varepsilon[/tex] of [tex]a[/tex]. (an open interval)

Given a set, [tex]A subset mathbb{R}[/tex] and an infinite sequence, [tex]{x_n }_{n=1}^{infty}[/tex], the sequence is said to be frequently in the set [tex]A[/tex] if for every[tex]N in mathbb{Z}^+[/tex], there exists some [tex]n geq N[/tex] such that [tex]x_n in A[/tex]. On the other hand, the sequence is said to be eventually in [tex]A[/tex], if there exists some [tex]N in mathbb{Z}^+[/tex] such that for every [tex]n geq N[/tex], we have [tex]x_n in A[/tex]. We can define convergence of a sequence as saying that a sequence converges to a point if it is eventually in every epsilon neighborhood of some real number.

TRANSLATION: To the non-mathematician, think of an infinite sequence simply as an infinite list of numbers. The sequence is frequently in a set, if no matter how far you go out in the list, there is another item in the list after that, which is in the set. On the other hand, the sequence is eventually in a set if you can go far enough out in the list so that all the elements after that point are in the set. The convergence of a sequence means that we can get arbitrarily close to some point by going far enough out in the list.

So, what is the spiritual principle?

I like to think of my life, more specifically, my faith development as a sequence of discrete moments where I demonstrate my faith in Christ. Throughout each day, I have opportunities to obey God’s call on my life or choose my own selfish desires. I involve myself in activities that can lead me closer to my Savior or further away. I think we’d all agree that there is NO guarantee in the Christian life that each day that goes by, NECESSARILY, draws us closer to God and to a godly character. That is, just because I am a Christian, doesn’t mean that I am better today than I was yesterday.

However, (and this is a big however), I believe that the work of the Holy Spirit is to shape us into the character of Christ, ** eventually**. Think of the character and life that God has designed you for. That is the point toward which our life is converging, once we have been saved. Now, we may oscillate near and far, but we will

*be closer. If you could measure the closeness to that point, we could say that given a level of character close to that of Christ, there is some point in time past which we will be that close. If you read Paul, in his letters, he often speaks of all the aspects of character to which we should apply ourselves, and he always does so with the expectation that we CAN obtain it. We are converging to the character of Christ.*

**eventually**Now, without grace, we may be * frequently *in the neighborhood of that point but we cannot converge. It is only through grace, that we may truly converge to the mind of Christ.

What do you think?

I can’t believe I’m saying this, but – that makes sense.

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She’s almost converted!!!

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I am glad God used math to make this point for you. The spiritual part makes sense to me–the math part, um, that calculus class I took as an undergradute is way too far in my distant past.

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So, will we ever actually make it???

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Paul said, “I am confident of this, He who began a good work in you will complete it in the day of Jesus Christ.” Philippians 1:6

Amen!

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