Show $n^{-1} \max_{k\leq n} X_k \rightarrow 0 \quad a.s.$ if and only if $\mathbb{E}(X_1)_+-\infty)>0$
46773
Using the first and second Borel-Cantelli Lemma to find necessary and sufficient condition for convergence in probability ($98\%$ solved)
110222
Maximal inequality for a sequence of partial sums of independent random variables
148582
Proof: $\|X\|_2\leq \sqrt{n}\|X\|_\infty$
35867
Squaring sides of an inequality
70926
Showing that for $n\geq 3$ the inequality $(n+1)^n<n^{(n+1)}$ holds
19804
How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
80873
Kind of converse of Kolmogorov maximal inequality
11465
Expected value of $\max\{X_1,\ldots,X_n\}$ where $X_i$ are iid uniform.
112510
Prove a.s. convergence of $(X_n)_n$ satisfying $E(X_{n+1} \mid F_n) \leq X_n+Y_n$ for $\sum_n Y_n<\infty$
43538
Show $\lim_{n \to \infty} n^{-1} E \left( \frac{1}{X}1_{[X>n^{-1}]} \right) =0$
8617
How do you prove ${n \choose k}$ is maximum when $k$ is $ \lceil \tfrac n2 \rceil$ or $ \lfloor \tfrac n2\rfloor $?
86635
Optimizing $(X_{n}+c)^{2}$ to show that $P(\overline{X}_{n}\geq \lambda) \leq E(X_{n}^{2})/(EX_{n}^{2}+ \lambda^{2})$
57778
Convergence in Distribution of the maximum of a sequence.
51347
Prove that $|a|\leq \max\{|b|,|c|\}$ if $b\leq a \leq c$
65662
Prove $\forall K > 0: \lim_{n\rightarrow\infty} \sqrt[n]{K} = 1$
133809
Inequality with sum of Binomial coefficients: $\sum\limits^n_{k=1}k \sqrt{\binom nk}\leq\sqrt{2^{n-1}n^3}$
95694
Is there a name for the function $\max(x, 0)$?