# 245A, Notes 4: Modes of convergence What's new

If one has a sequence \$latex {x_1, x_2, x_3, ldots in {bf R}}&fg=000000\$ of real numbers \$latex {x_n}&fg=000000\$, it is unambiguous what it means for that sequence to converge to a limit \$latex {x in {bf R}}&fg=000000\$: it means that for every \$latex {epsilon > 0}&fg=000000\$, there exists an \$latex {N}&fg=000000\$ such that \$latex {|x_n-x| leq epsilon}&fg=000000\$ for all \$latex {n > N}&fg=000000\$. Similarly for a sequence \$latex {z_1, z_2, z_3, ldots in {bf C}}&fg=000000\$ of complex numbers \$latex {z_n}&fg=000000\$ converging to a limit \$latex {z in {bf C}}&fg=000000\$.

More generally, if one has a sequence \$latex {v_1, v_2, v_3, ldots}&fg=000000\$ of \$latex {d}&fg=000000\$-dimensional vectors \$latex {v_n}&fg=000000\$ in a real vector space \$latex {{bf R}^d}&fg=000000\$ or complex vector space \$latex {{bf C}^d}&fg=000000\$, it is also unambiguous what it means for that sequence to converge to a limit \$latex {v in {bf R}^d}&fg=000000\$ or \$latex {v in {bf C}^d}&fg=000000\$; it means…

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