I got an example exam in front of me and the first task has got several short statements where you need to say "true" if true and "false" if statement wrong (no need to say reason).
a) The series $\sum_{n=0}^{\infty}a^{k}$ is convergent for all $a \in \mathbb{R}$ with $a>0$.
b) The sequence $(x_{n})_{n \in \mathbb{N}}$ with $a_{n}=\frac{1}{n}$ if $n \in \mathbb{N}$ odd and $a_{n}=1$ if $n \in \mathbb{N}$ even has only got one cluster point.
c) $\inf \left \{x\in\mathbb{R}|x\geq0 \wedge e^{x}<1\right \}=0$
a) False. Let $a$ and or $k$ be $\infty$ then the series will diverge.
b) False because the sequence $a_{n}=\frac{1}{n}$ has already got two cluster points (one at $1$ and the other at $0$).
c) True. First condition is clear that $0$ will be the smallest value. For the second condition, the only way getting smaller values than $1$ is by inserting negative values for $x$. Thus we will always have $\frac{1}{e^{x}}$ where $0$ is the infimum as well.
What do you think about this? Are my answers correct? Are my justifications correct?