List questions from formal-proofs

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For $a$ is a real number, suppose $(x_n)$ is a sequence such that for all $n$, $|x_n - a| < 1/n$. Prove $lim x_n = a$?

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Suppose $(s_n)$ is a sequence such that $\lim_{n \to \infty} s_n = 7$ and $s_n < 7$ for all $n\in \Bbb N$.

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help verifying my answer for this" premise-conclusion" question

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Orthogonal Vectors and Projections

Use rules of inferential logic for the following problem..

Show that $(p \to q) \lor (q \to p)$ is a tautology

$p \to (q\vee\neg r), \neg q, r ⊢ \neg p$ - Natural deduction- elimination with $\neg$ operator

Should $x$ be not free in $\beta$ to prove $\vdash [ \forall x(\beta\rightarrow \alpha)\rightarrow (\exists x\beta\rightarrow \alpha)]$?

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For $a$ is a real number, suppose $(x_n)$ is a sequence such that for all $n$, $|x_n - a| < 1/n$. Prove $lim x_n = a$?

Suppose $(s_n)$ is a sequence such that $\lim_{n \to \infty} s_n = 7$ and $s_n < 7$ for all $n\in \Bbb N$.

help verifying my answer for this" premise-conclusion" question

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Proving using axioms of propositional logic

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