$\sum_{k=1}^{\infty}|a_kb_k| < \infty$ $\forall \sum_{k=1}^\infty|a_k|$ convergent. Prove $\{b_k\}_{k=1}^\infty$ is bounded. For continuous functions?
113384
$\sum_{n\geq1} k_n x_n$ converges for every $x \in l_1$, then $\sup_{n\geq1} |k_n| < \infty$
23894
$a_n>0$ and $(a_n)$ is decreasing. Suppose that $\sum_{n=1}^{\infty}{a_{2n}}$ converges. Prove that $\sum_{n=1}^{\infty}{a_n}$ also converges.
98247
Is $ \sum\limits_{n=1}^\infty \frac{|\sin n|^n}n$ convergent？
97968
Does $\big|\sum_{k=1}^na_k\big|\leq\sqrt{n}$ imply the convergence of $\sum_{1}^\infty \frac{a_k}{k}$?
29836
Prove: Convergent sequences are bounded
131425
Prove that the convergence of $\prod_{n=1}^{\infty} b_n$ implies $\lim_{n→∞} b_n = 1$
55914
If $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent
20142
$\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$
124615
Proving convergence of sums: If $\sum_{n=1}^{\infty}a^2_n$ and $\sum_{n=1}^{\infty}b^2_n$ converge then $\sum_{n=1}^{\infty}a_nb_n$ converges
38185
Prove that $\{\cos(nx) : n=1, ... ,\infty \}$ is bounded but not totally bounded in $C([0,\pi])$
26500
Prove $\sum_{n=1}^{\infty}|a_{n}b_{n}|$ converges if $\sum_{n=1}^{\infty}a_{n}^{2}$ and $\sum_{n=1}^{\infty}b_{n}^{2}$ converge
44706
Behavior of $\sum a_n x^n$ given $\sum |a_n - a_{n-1}| < \infty$
129087
Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$
74501
$\delta : C([0,1])\to\mathbb{R}$, $f\mapsto f(0)$, with sup-norm is bounded but unbounded with one-norm.
129393
Product of bounded and convergent to $0$ sequence is a convergent to $0$ sequence
121079
lemma: Cauchy sequences are bounded.
155475
$\lim\limits_{n\to \infty} \int_0^1 |f(x)-a_nx-b_n| dx=0$ implies $(a_n)_n,(b_n)_n$ are convergent