Hello fellow mathematicians I hope that the summer hasn't rusted your brains here is a question:

Let be a sequence and .

Suppose that and both converge to .

Prove that also converges to L.

Is this the correct way of doing this? I'm quite rusty at the moment and any help would be appreciated.

We know the following to be true:

Let and . since

- such that
- such that

So we get:

which after replacing variables and moving the inequality, is equivalent to :

- such that
- such that

Let . Choose . Let . .

- such that
- such that

Is this proof any good? I don't really know I kind of just scribbled it down but if there is anythign wrong I would like to know thanks for the help