Why does the following sentence end in the て form?


In a conversation between a student (Mike) and his teacher:

> 先生: ええ。[子供の]頃、私の家の近くに柔道をやっていてとても元気な子供がい **て** ね。

the latter ends the sentence with the ~て form. Why is that so? I would expect
the sentence to end with いました rather than いて, since he is speaking about a
past time and he is ending all the other sentences with the polite marker です.

I found a related [question](https://japanese.stackexchange.com/q/27651/32952)
with its [answer](https://japanese.stackexchange.com/a/27665/32952) about
another sentence ending in ~て in a different context. In that case, the ~て
ending is used to "imply that the speaker has something more to say". I'm not
sure if it's also the case in the sentence we are discussing here, because the
conversation continues after that. If this is another case of a sentence where
it's implied that there's more to say, what would that extra information be?
And if it's not the case, what else is going on here?

For your reference, this is the full conversation:

[![enter image description
here](https://i.stack.imgur.com/Ivjr6.jpg)](https://i.stack.imgur.com/Ivjr6.jpg)

よろしくお願いします




At first, I wanted to answer like this. When the teacher is about forty years
old and the student is about fifteen, the teacher will not use the polite form
of 子供がいましてね but 子供がいてね. But when the student is twenty-five, the teacher will
probably use 子供がいましてね.

But after reading the full conversation, I have found マイク using the very
polite form of speech such as やっていらっしゃったそうですね. This kind of speech form makes
me guess that マイク will be over twenty.

My answer is that this sentence sounds a little awkward among other sentences
of polite form, or does not belong properly to the whole conversation.

