$f_n, f \in L^1$ and pointwise convergence implies ...
9613
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$
52711
Prove that the sequence $\{v_n\}$ converges to the maximum value of $f$ on $[a,b]$
84818
Convergence of $\sum_{n=1}^{\infty}x_n\overline{y_n}$ for every $(x_n)\in c_0$ implies $\sum_{n=1}^{\infty}|y_n|<\infty$
66198
Sufficient condition for convergence of a real sequence
119348
Find sequence of differentiable functions $f_n$ on $\mathbb{R}$ that converge uniformly, but $f'_n$ converges only pointwise
113384
$\sum_{n\geq1} k_n x_n$ converges for every $x \in l_1$, then $\sup_{n\geq1} |k_n| < \infty$
153063
Uniform convergence of $f_n(x)=nx^n(1-x)$ for $x \in [0,1]$?
40349
Continuity of analytic function implies convergence of power series?
26126
$f \in L^1$, but $f \not\in L^p$ for all $p > 1$
48500
Proof of convergence of sequences
44057
Pointwise but not uniform convergence of continuous functions on $[0,1]$
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
76593
Show that $f_n(z)=z^n$ converges locally uniformly on $|z|<1$ but not uniformly(?)
2239
Showing that $\{f_n \}$ converges to $f$ is equivalent to $\lim_{n\to \infty} \int_X \frac{|f_n(x)-f(x)|}{1+|f_n(x)-f(x)|}d\mu(x)=0$
19858
Proving $f$ is $L^1$ implies convergence of $\sum_{k \in \mathbb{Z}}{ 2^k*m(|f| > 2^k)}$
60556
Uniform convergence, Bounded derivative.
117649
$f:[0,1]\to[0,1]$ be a continuous function. Let $x_1\in[0,1]$ and define $x_{n+1}={\sum_{i=1}^n f(x_i)\over n}$.Prove, $\{x_n\}$ is convergent