# Two questions on sequences

These are homework questions I was assigned to do:

1. Let $(x_{n})_{n \geq 0}$ be a convergent sequence in $\mathbb{R}$ with limit $x$ and let $a,b \in \mathbb{R}$ with $a \leq b$. Show that if $a \leq x_{n} \leq b$ for all $n$, then it is also true that $a \leq x \leq b$.

I tried proving this by means of the triangle inequality (both the "regular" theorem and its reverse) and I tried finding a proof by contradiction, but to no avail. Could you help me out?

2. Let $(x_{n})_{n \geq 0}$ be a convergent sequence and $(y_{n})_{n \geq 0}$ be a divergent sequence. Show that $(x_{n} + y_{n})_{n \geq 0}$ diverges.

I tried proving this with the definition of both convergent and divergent sequences, assuming that $(x_{n} + y_{n})_{n \geq 0}$ converges and then I tried to find a contradiction, but I couldn't find one.

Max Muller

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Contradiction seems like the way to go. Suppose $x_n = c = b + \gamma > b$. The limit exists, so there is a $\epsilon$ ball for some $N$ so that all the terms of the sequence are in it. But if $\epsilon < \gamma/2$, say, we have a contradiction, as no $x_n > b$.
For the second one, you can go straight from the definition. Since the convergent converges to $L$, say, consider $N$ so that the divergent sequence gets bigger than $M - |L|$. It diverges, each of these exist, and so it still diverges.