These are homework questions I was assigned to do:
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?
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.
Thanks in advance,
Max Muller